LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA 


GIFT  OF 

4j U3 B 


ouuuin. 


Ctes 


THE 


GENERAL   PRINCIPLES 


OF 


PHYSICAL      SCIENCE 

AN    INTRODUCTION    TO    THE    STUDY    OF   THE 
GENERAL    PRINCIPLES    OF    CHEMISTRY 


BY 


ARTHUR   A.    NOYES 


PROFESSOR  OF  THEORETICAL  CHEMISTRY  IN   THE   MASSACHUSETTS 
INSTITUTE  OF  TECHNOLOGY 


NEW  YORK 
HENRY    HOLT    AND    COMPANY 

1902 


KAPH   MIM 
GHKM.BLDG,     U.  C 


COPYRIGHT,  1902, 
BY  ARTHUR  A.  NOYBS. 


TO 

PROFESSOR  WILHELM  OSTWALD 

TO  WHOSE  INSPIRING  TEACHING  THE  AUTHOR 

OWES  HIS  FIRST  ACTIVE  INTEREST  IN 

GENERAL    CHEMISTRY 

THIS  BOOK  IS  DEDICATED. 


111074 


PREFACE. 

As  the  title  of  this  book  indicates,  its  purpose  is  to 
present  the  general  concepts  and  laws  of  physics  and  chem- 
istry which  lie  at  the  basis  of  the  modern  science  of  theo- 
retical chemistry.  It  forms  the  first,  introductory  part  of  a 
projected  work  upon  this  science,  to  be  entitled  the  General 
Principles  of  Chemistry,  the  later  parts  of  which  are  to  treat, 
in  succession,  of  the  General  Theories  of  Chemistry,  of  the 
Relations  between  Physical  Properties  and  Chemical  Com- 
position, of  the  Principles  relating  to  the  Occurrence  and 
Equilibrium  of  Chemical  Changes,  and  of  the  Principles  re- 
lating to  the  Energy-changes  Attending  Chemical  Changes. 
As  the  work  has  had  to  be  discontinued,  it  has  been  thought 
advisable  to  publish  the  part  already  completed,  in  the  hope 
that  it  may  assist  students  of  theoretical  chemistry  by  supply- 
ing, in  a  concise  and  consistent  form,  the  essential  preliminary 
knowledge  of  the  fundamental  principles  of  physical  science. 
The  method  of  presentation  of  these  principles  may  also  be 
of  interest  to  teachers  of  general  physics  and  chemistry,  since 
an  effort  has  been  made  to  attain  precision  in  the  statement 
of  laws  and  definitions,  since  the  energy-concept  has  been 
employed,  as  far  as  seemed  possible,  as  the  basis  of  the  con- 
sideration of  other  physical  concepts,  and  since  the  difficult 
subject  of  the  Second  Law  of  Energetics  has  been  very  fully 
discussed  from  a  non-mathematical  standpoint. 

The  treatment  throughout  is  a  systematic,  not  a  histori- 
cal one.  It  is  essentially  non-mathematical,  but  it  is  assumed 
that  the  reader  knows  the  significance  of  differential  and  in- 
tegral expressions,  and  that  he  is  acquainted  with  a  few  of  the 
simpler  operations  of  the  Calculus.  On  the  purely  physical 
side,  the  treatment  is  complete  and  intelligible  in  itself,  in 
the  sense  that  each  term  employed  is  first  defined ;  but  it 
has  been  necessary  to  so  condense  it,  that  it  can  hardly 
be  fully  appreciated  except  by  one  who  has  had  a  fairly 


vi  PREFACE. 

thorough  course  in  general  physics  :  on  this  side,  the  book 
is  intended  to  serve  as  a  systematic  review  of  the  important 
physical  concepts  and  principles,  and  as  an  aid  in  acquir- 
ing a  definite  and  precise  conception  of  them.  It  is  the 
practice  of  the  author  to  emphasize  their  significance  by 
requiring  of  his  classes  the  solution  of  numerous  prob- 
lems. On  the  other  hand,  those  considerations  which 
have  an  especially  important  chemical  bearing  are  very 
fully  presented.  This  is  true,  for  example,  of  the  charac- 
teristics of  chemical  substances,  of  the  concepts  of  combin- 
ing, equivalent,  and  molecular  weights,  of  the  physical 
properties  and  energy-relations  of  gases,  of  the  work  involved 
in  volume-changes,  of  Faraday's  Law,  and  of  the  First  and 
Second  Laws  of  Energetics.  Theories  are  not  discussed  at 
all  in  this  book ;  for,  in  the  author's  opinion,  it  is  desirable, 
in  order  to  avoid  producing  an  exaggerated  idea  of  their 
significance,  to  present  fundamental  principles  independently 
of  them. 

References  are  not  inserted  in  the  body  of  the  book, 
but  are  given  in  an  appendix.  Their  purpose  is  to  bring 
to  the  attention  of  the  reader  more  extended  discussions 
of  the  subjects  treated  in  the  text,  to  give  the  authorities 
for  the  numerical  data  cited,  and  to  indicate  the  works  to 
which  the  author  is  especially  indebted.  The  notation  em- 
ployed is  summarized  in  a  second  appendix  ;  for  much  atten- 
tion has  been  devoted  to  this  matter,  with  the  view  of 
devising  for  physico-chemical  purposes  a  system  consistent  in 
itself  and  yet  according  as  nearly  as  possible  with  common 
usage. 

In  conclusion,  the  author  wishes  to  express  his  great  in- 
debtedness to  many  friends  for  criticisms  and  suggestions,  es- 
pecially to  Dr.  H.  M.  Goodwin,  Prof.  H.  E.  Clifford,  Dr.  W. 
D.  Coolidge,  Dr.  E.  Weintraub,  Mr.  J.  G.  Coffin,  and  Mr.  M. 
Rosenberg. 

ARTHUR  A.  NOYES. 

Boston,  June,  igoa. 


CONTENTS. 

CHAPTER  I.     THE    OBJECT,    THE    METHODS,    AND  THE 
SUB-DIVISIONS  OF  SCIENCE. 

FAGB 

1.  The  Object  of  Science      -        -       »       -       -       ^     •-  3 

2.  The  Methods  of  Science  -        -     "  ~       -  '    -    -    -       -  4 

3.  The  Subdivisions  of  Science    -        -        -        -       •'-•'.  9 

CHAPTER  II.    THE  FUNDAMENTAL  CONCEPTS  OF  PHYS- 
ICAL SCIENCE. 

4.  The  Fundamental  Concepts 10 

5.  Space  and  Time 10 

6.  The  Concepts  of  Matter  and  Energy  11 

CHAPTER  III.    THE    GENERAL    PRINCIPLES    RELATING 
TO  MATTER. 

7.  Matter,  its  Quantitative  Measurement,  and  the  Law 

of  its  Conservation 14 

8.  The  States  of  Aggregation  and  Other  Physical  States 

.of  Matter       -       -       -       -       -!      -       -       .       -  19 

9.  Chemical  Substances  and  Mixtures        -       -       -       -  28 

10.  Elementary  and  Compound  Substances          »      .-    ,  -  32 

11.  Law  of  the  Conservation  of  the  Elements      «•       -       •-  33 

12.  The  Law  of  Definite  Proportions    -        -       ...  35 

13.  The  Law  of  Multiple  Proportions  -        -       -       -       -  37 

14.  The  Law  of  Combining  Weights    -        -       -       .      ...  38 

15.  Determination  of  the  Combining  Weights   ,-      ..  -        -  42 

16.  Numerical  Values  of  the  Combining  Weights  45 

17.  Elementary  Composition  as  a  Means  of  Distinguishing 

Chemical  Substances  from  Mixtures  48 

18.  Chemical  Formulas  and  Chemical  Equations  49 

19.  Definition  of  Equivalent  Weights   -       -       -   „    -       -  52 

20.  General  Significance  of  the  Properties  of  Gases  56 

21.  Relation  between  the  Pressure  and  Volume  of  Gases. 

Boyle's  Law 56 

22.  Relation  between  the  Pressure-Volume  Product  and 

Temperature.    Gay-Lussac's  Law  of  Temperature- 
Effect     61 

23.  Relation  between  the  Pressure- Volume  Product  and 

Combining   Weight.      Gay-Lussac's   Law   of    Com- 
bining Volumes    -      —       •       -       -       ...  64 

24.  General  Expression  of  the  Pressure-Volume  Relations 

of  Gases.    Empirical  Definition  of  Molecular  Weight  65 

vii 


viu  CONTENTS. 

CHAPTER  IV.    THE    GENERAL   PRINCIPLES   RELATING 
TO  ENERGY. 

PACK 

25.  The  Forms  of  Energy  and  Other  Classes  of  Energy 

Manifestations       -       -       -       -       ...       _  69 

26.  The  Quantitative  Measurement  of  Energy  74 

27.  The  Law  of  the  Conservation  of  Energy,  or  the  First 

Law  of  Energetics        - . '•-   "    -       -       -       -       -  76 

28.  The  Factors  of  Energy  in  General  -       -   .  •; ».    —       -  79 

29.  The  Factors  of  Kinetic  and  Gravitation  Energies.    De- 

finitions of  Force          -       -       -       -       -       -       -  82 

30.  The  Factors  of  Surface,  Volume,  and  Elastic  Energies  88 
81.    Electricity  and  Magnetism.     Coulomb's  Law.    Electric 

Currents        ------...99 

32.  The  Factors  of  Electrical  Energy.     Ohm's  Law  and 

Joule's  Law  -       -      ;-:       -       -       -       -       -       -  107 

33.  Faraday's  Law  of  Electrolytic  Conduction    -       -       -  114 

34.  Heat  Energy    -       -       -       -       •       -        -        -        -  120 

35.  Chemical  Energy      -        ...       .       .        -        -  124 

36.  Radiant  Energy 125 

37.  The  Internal  Energy  of  Gases.    Experiments  of  Gay- 

Lussac  and  of  Joule  and  Thomson      -  132 

38.  The  Second  Law  of  Energetics                                        -  137 

39.  Application  of  the  Second  Law  to  Changes  Taking 

Place  at  a  Constant  Temperature        -  139 

40.  Application  of  the  Second  Law  to  Changes  Taking 

Place  at  Different  Temperatures                                -  148 

APPENDIX    I.     REFERENCES       -       -       ....  164 

APPENDIX  II.    NOTATION     ...       -       -       -       -  166 

INDEX     ,-                               168 


CHAPTER   I. 

THE   OBJECT,    THE   METHODS,   AND    THE    SUBDIVISIONS    OF 

SCIENCE. 

i.  The  Object  of  Science.  — It  is  the  object  of  sci- 
ence to  facilitate  the  acquirement  of  a  knowledge  of  the 
phenomena  of  nature  by  devising  means  for  their  mental 
representation,  thus  replacing  by  simple  operations  of 
thought  the  slow,  laborious  process  of  acquiring  that 
knowledge  by  the  observation  of  the  innumerable  isolated 
phenomena.  //  is  therefore  the  object  of  science  to  make  the 
completest  possible  presentation  of  natural  phenomena  in  such 
a  manner  that  a  knowledge  of  them  can  be  acquired  with  the 
least  possible  expenditure  of  effort. 

It  is  to  be  noted  that  the  acquirement  of  this  knowl- 
edge has  two  distinct  aspects :  on  the  one  hand,  science 
aims  to  make  as  easy  as  possible  the  comprehension  on  the 
part  of  individuals  of  the  knowledge  already  acquired  by 
mankind  ;  and,  on  the  other,  it  aims  to  add  to  the  total 
sum  of  human  knowledge  by  leading  to  the  discovery  of 
previously  unknown  phenomena. 

This  statement  of  the  object  of  science  should  be  care- 
fully noted,  in  order  that  the  reader  may  fully  appreciate 
the  significance  of  the  various  principles  and  hypotheses 
employed  for  the  mental  representation  of  phenomena,  it 
being  evident  that  the  value  of  such  means  of  representa- 
tion is  to  be  estimated  solely  through  a  consideration  of  the 
extent  to  which  they  assist  in  attaining  that  object  —  that 
is,  in  facilitating  the  acquirement  of  a  knowledge  of  natural 
phenomena. 

It  will  be  clear  from  this  statement  that,  however 
abstract  or  theoretical  the  methods  of  science  may  be,  its 
object,  the  saving  of  labor  in  the  acquirement  of  knowledge, 
is  a  concrete  and  highly  practical  one.  Moreover,  the 


4  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

knowledge  which  it  is  the  ultimate  object  of  natural  science 
to  make  readily  available,  is  only  the  knowledge  of  actual 
phenomena,  that  is,  of  those  natural  conditions  and  changes 
which  lie  within  the  possible  range  of  our  experience. 
Thus,  it  is  not  a  part  of  its  ultimate  object  to  acquire 
knowledge  in  regard  to  mentally  conceived  existences,  such 
as  the  atoms  of  matter,  or  the  particles  of  luminiferous 
ether,  which  are  of  such  a  magnitude  and  character  as  to 
lie  far  beyond  the  limits  of  human  perception.  By  this 
statement  it  is  not,  however,  meant  to  deny  that  such 
mental  conceptions  may  be  of  great  scientific  value,  or  that 
the  development  of  them  forms  a  legitimate  part  of  the 
process  by  which  the  object  of  science  is  to  be  attained ; 
but  it  is  desired  to  emphasize  the  idea  that  such  knowledge 
is  the  means  to  an  end,  and  not  the  end  itself. 

The  value  of  a  knowledge  of  natural  phenomena  is  so 
generally  recognized  that  it  is  not  necessary  to  discuss  it. 
But,  in  order  to  guard  against  a  misinterpretation  of  the 
statement  that  the  object  of  science  is  a  practical  one,  by 
which  was  meant  that  the  knowledge  aimed  at  is  only  that 
of  actual  phenomena,  it  may  be  added  that  the  value  of 
science  arises  from  the  fact  that  it  satisfies  the  demands 
of  the  intellectual  faculties  of  man  —  his  desire  to  know 
the  phenomena  of  the  external  world  with  which  he  is  in 
contact  —  as  well  as  from  the  fact  that  it  greatly  assists 
in  promoting  his  material  welfare. 

2.  The  Methods  of  Science.  —  The  first  stage  in  the 
development  of  a  science  is  the  acquirement  by  observation 
and  experiment  of  a  knowledge  of  some  of  the  specific 
phenomena  of  nature.  Some  of  the  isolated  facts  which 
it  will  be  the  object  of  the  science  to  correlate,  and  thus 
make  more  readily  comprehensible,  are  first  accumulated. 
These  constitute  the  data  of  the  science. 

For  the  correlation  of  these  facts  two  distinct  methods 
are  employed,  known  respectively  as  the  inductive  or  empiri- 
cal method  and  the  deductive  or  theoretical  one. 


THE  METHODS  OF  SCIENCE.  5 

The  inductive  method  consists  in  a  comparison  of  the 
isolated  observations  with  one  another,  with  the  object  of 
deriving  from  them  some  general  statement  which  shall  em- 
brace and  sum  up  a  number  of  the  phenomena  or  of  their 
characteristics.  A  general  statement  or  principle  of  this 
kind  so  derived  is  called  an  induction,  and  the  process  of 
reasoning  by  which  it  is  derived  is  called  inductive  reason- 
ing; that  is,  inductive  reasoning  is  the  mental  process  by 
which  general  principles  are  derived  from  a  consideration 
of  specific  cases.  An  induction  which  has  been  derived  by 
a  consideration  of  a  comparatively  few  specific  phenomena, 
must  next  be  tested  as  to  the  extent  of  its  applicability 
to  phenomena  in  general.  If  it  is  found  that  the  principle 
is  one  which  expresses  a  common  characteristic  of  a  large 
number  of  natural  phenomena,  and  that  it  is  not  in  contra- 
diction with  any  phenomenon,  it  is  designated  an  empirical 
law  of  nature. 

It  is  important  to  realize,  however,  that  the  universal 
validity  of  any  such  so-called  law  of  nature  is  always  a  ques- 
tion of  greater  or  less  probability ;  for,  since  it  is  utterly 
impossible  to  test  the  law  by  applying  it  to  all  phenomena, 
it  is  never  certain  that  some  case  will  not  be  discovered  in 
which  the  law  does  not  hold  true.  The  probability  that  a 
law  will  prove  to  be  applicable  to  any  new  phenomenon  is 
of  course  greater,  the  greater  the  number,  and  the  more 
varied  the  character,  of  the  phenomena  to  which  it  has  been 
already  found  applicable  ;  and  only  when  these  phenomena 
are  extremely  numerous  and  diversified  does  the  probabil- 
ity that  the  law  is  universally  true  approach  a  certainty. 

In  the  application  of  an  accepted  law  to  a  new  phe- 
nomenon the  scientist  must  therefore  always  consider  the 
character  and  extent  of  the  experience  which  has  led  to  its 
acceptance  :  he  must  not,  on  the  one  hand,  underestimate, 
as  unscientific  minds  are  apt  to  do,  the  probability  that  a 
well-established  law  will  hold  true  in  the  case  of  any  new 
phenomenon ;  nor  must  he,  on  the  other,  dogmatically  assert 


6  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

that  the  alleged  phenomenon  is  impossible  because  it  is 
inconsistent  with  the  accepted  law.  For  example,  the  laws 
that  neither  matter  nor  energy  is  ever  created  or  destroyed 
are  based  on  so  extensive  and  varied  an  experience  in  all 
branches  of  science,  that  a  statement  that  a  process  for  the 
creation  of  matter  has  been  discovered  or  a  perpetual  motion 
machine  invented  is  so  improbable  as  scarcely  to  deserve 
scientific  consideration.  The  law  that  certain  substances, 
the  so-called  elements,  are  not  transformed  into  one  another 
by  any  process  whatever,  is  likewise  one  to  which  a  very 
high  degree  of  probability  attaches ;  for  the  large  number 
of  attempts  that  have  been  made  to  effect  such  a  transfor- 
mation have  all  resulted  in  failure.  Nevertheless,  a  consid- 
eration of  the  character  of  the  evidence  in  favor  of  the 
universal  validity  of  this  law  leads  to  the  conclusion  that 
the  probability  of  discovering  an  exception  to  it,  although 
slight,  is  considerably  greater  than  that  of  meeting  with  an 
exception  to  the  Laws  of  the  Conservation  of  Matter  and 
Energy ;  a  statement  that  one  of  the  well-established  ele- 
ments had  been  transformed  into  another  should,  therefore, 
before  it  is  accepted,  be  subjected  to  the  closest  scientific 
criticism,  and  confirmed  by  conclusive  experimental  proof. 

The  discovery  that  a  law  is  not  universally  applicable 
does  not  entirely  destroy,  though  it  does  lessen,  its  sci- 
entific value :  it  only  increases  the  probability  that  other 
exceptions  to  the  law  will  be  met  with.  It  is  in  such  a 
case  desirable  to  establish  definitely  by  experimental  inves- 
tigation the  class  of  phenomena  to  which  the  law  invari- 
ably applies;  for,  though  restricted  thereby  in  scope,  the 
degree  of  certainty  attaching  to  applications  of  it  is  greatly 
increased. 

It  is  evident  that  the  object  of  science  might  be  com- 
pletely attained  by  the  inductive  method  if  it  were  possible 
to  derive,  from  an  examination  of  the  data,  a  few  principles 
of  such  comprehensiveness  and  definiteness  that  the  vast 
number  of  natural  phenomena  could  be  represented  by 


THE  METHODS  OF  SCIENCE.  7 

them.  Experience  has  shown,  however,  that,  on  account 
of  the  great  complexity  of  phenomena,  it  is  not  possible  — 
at  any  rate,  immediately  —  to  derive  inductively  universal 
principles  of  this  character.  Recourse  is  therefore  had  to 
the  second  method  referred  to  above:  failing  to  effect 
further  simplification  through  a  consideration  of  the  facts 
themselves  and  of  the  laws  already  derived  from  them,  in- 
vestigators attempt  to  accomplish  it  by  imagining  some 
state  of  things  which  has  some  of  the  phenomena,  or  of 
the  laws  generalizing  them,  as  its  necessary  consequences. 
A  concept  of  this  kind,  of  which  no  direct  experimental 
verification  has  been  given,  is  called  a  hypothesis ;  that  is, 
a  hypothesis  is  an  assumption  of  the  existence  of  conditions 
of  which  we  have  no  direct  experimental  evidence,  having 
for  its  purpose  the  correlation  of  known  phenomena  and 
the  discovery  of  new  ones.  A  hypothesis  of  this  kind  hav- 
ing been  formed,  all  its  logical  consequences  are  derived 
from  it,  and  the  conclusions  thus  reached  are  tested  by 
observation  and  experiment  as  to  their  correspondence  with 
actual  phenomena. 

The  process  of  reasoning  by  which  such  conclusions 
are  derived  is  called  deductive  or  a  priori  reasoning ;  that  is 
to  say,  deduction  is  the  mental  process  by  which  special  con- 
clusions are  derived  from  more  general  propositions.  It  is 
the  inverse  of  induction.  Conclusions  which  have  been 
thus  derived  from  hypotheses,  but  have  received  no  direct 
experimental  proof,  will  be  designated  theoretical  principles 
throughout  this  work,  the  term  law  being  confined  to  gen- 
eral statements,  whether  arrived  at  inductively  or  deduc- 
tively, which  have  been  directly  verified  by  experiment. 
The  whole  body  of  conclusions  derived  from  a  single 
hypothesis  or  a  few  related  hypotheses,  constitutes  a 
theory.  The  larger  the  number  of  otherwise  unrelated  facts 
or  empirical  laws  which  the  theory  serves  to  correlate,  and 
the  simpler  its  character,  the  more  valuable  and  efficient  it 
is  as  a  means  of  simplification.  Aside  from  its  function  as 


«  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

a  means  of  facilitating  the  acquirement  of  existing  knowl- 
edge, a  theory  may  also  be  of  value  by  leading  to  the  dis- 
covery of  new  facts  and  new  laws. 

It  should  be  distinctly  recognized,  however,  that  the 
•development  of  hypotheses  and  theories  is  one  of  the 
methods  employed  in  attaining  the  object  of  science,  and 
not  the  object  itself,  and  that  they  are  to  be  discarded  when 
any  simpler  means  of  attaining  that  object  is  discovered. 
It  is  highly  important  that  the  student  of  science  should 
constantly  keep  in  mind  the  radical  distinction  between 
facts  and  laws  on  the  one  hand,  and  hypotheses  and  theo- 
retical principles  on  the  other ;  for  the  history  of  science 
and  education  proves  that  there  is  a  great  tendency  to 
attribute  to  the  latter  an  undue  importance,  so  that  a 
theory,  especially  one  which  has  already  proved  of  great 
scientific  value,  may  come  to  be  a  hindrance  to  increase  of 
knowledge  and  to  further  progress  of  the  science,  by  caus- 
ing facts  inconsistent  with  the  theory,  or  not  comprehended 
by  it,  to  be  ignored,  thus  giving  to  the  science  a  one-sided 
development.  It  is  also  a  very  common  error  to  believe 
that  the  knowledge  of  a  group  of  phenomena  or  the  value 
of  an  empirical  law  is  increased  merely  by  devising  a 
hypothesis  of  which  the  phenomena  or  law  can  be  shown 
to  be  a  necessary  consequence.  Thus,  by  reason  of  the 
fact  that  no  definite  hypothesis  in  regard  to  the  nature  of 
electricity  exists,  it  is  apt  to  be  thought  that  the  knowledge 
of  electrical  phenomena  is  vague  and  unsatisfactory,  which 
is  by  no  means  the  case.  If  a  theory  were  devised  which 
brought  to  light  new  relationships,  not  previously  recog- 
nized, between  electrical  phenomena  it  would,  indeed,  be 
of  scientific  value.  If,  without  doing  this,  it  merely  "ex- 
plained" a  law  relating  to  those  phenomena  by  showing  it 
to  be  the  consequence  of  a  certain  assumption,  it  might,  if 
adopted  seriously  as  a  part  of  the  science,  be  an  encum- 
brance to  it,  not  an  aid ;  for  a  hypothetical  element  which 
might  lead  to  false  conclusions  would  be  introduced,  with- 
out a  compensating  advantage.  If  it  be  fully  appreciated 


THE  SUBDIVISIONS  OF  SCIENCE.  9 

that  the  object  of  science  is  to  make  known  the  phenomena 
of  nature,  and  not  to  answer  the  question  why  they  are  so, 
there  will  be  less  danger  of  attributing  an  undue  weight  to 
hypothetical  considerations. 

3.  The  Subdivisions  of  Science.  —  For  the  sake  of 
convenience,  the  vast  number  of  natural  phenomena  with 
which  science  as  a  whole  has  to  deal  are  classified  into 
several  distinct  groups,  which  can  be  treated  separately  to 
a  certain  extent.  The  systematized  knowledge  relating  to 
these  groups  of  phenomena  constitutes  the  separate  sciences. 
These  may  be  primarily  subdivided  into  three  classes :  first, 
the  abstract  sciences,  logic  and  mathematics,  which  treat, 
qualitatively  and  quantitatively  respectively,  of  the  necessary 
relations  between  phenomena  —  that  is,  of  the  relations 
which  are  common  to  all  phenomena,  and  have  been  so  im- 
pressed on  the  mind  by  universal  experience  that  they  have 
become  conditions  of  thought ;  second,  the  abstract-concrete 
or  physical  sciences,  chemistry  and  physics,  which  treat  of 
the  properties  of  substances  and  bodies  in  the  abstract  — 
that  is,  without  reference  to  any  definite  aggregates ;  and 
third,  the  concrete  sciences,  which  treat  of  the  properties 
of  definite  aggregates,  for  example,  astronomy,  geology, 
biology,  and  anthropology,  which  treat  of  the  heavenly 
bodies,  the  earth,  living  bodies,  and  man,  respectively. 

The  provinces  of  the  two  physical  sciences,  chemistry 
and  physics,  may  be  defined  as  follows.  Chemistry  treats 
of  the  specific  properties  of  chemical  substances,  both  in  the 
pure  state  and  as  components  of  mixtures,  and  of  their  trans- 
formations into  one  another.  Physics  treats  of  the  proper- 
ties of  bodies  without  reference  to  the  chemical  substances 
contained  in  them,  and  of  all  changes  except  those  involving 
transformations  of  chemical  substances.  The  terms  used  in 
these  definitions  are  explained  in  sections  6  and  9. 

This  Part  of  the  book  is  devoted  to  the  fundamental 
considerations  underlying  both  of  these  sciences.  It  has 
therefore  been  entitled  The  General  Principles  of  Physical 
Science. 


CHAPTER   II. 

THE    FUNDAMENTAL   CONCEPTS    OF    PHYSICAL   SCIENCE. 

4.  The  Fundamental  Concepts. — The  study  of  natu- 
ral phenomena  has  led  to  the  adoption  of  four  most  funda- 
mental concepts,  or  abstract  ideas,  called  space,  time,  matter, 
and   energy.     As  some,  or  all,  of   these   concepts  are   in- 
volved   in    the    explanation    of    every   phenomenon,    their 
characteristics,   the   laws  relating  to  them,  and   the  units 
employed  for  their  measurement  will  be  first  considered. 

5.  Space  and  Time.  —  The   concepts   of   space   and 
time  have  become  familiar  to  us  from  continual  experience. 
They  are  so  fundamental  and  abstract  in  character  that  any 
definition  of   them  would  be  only  a  substitution  of  terms 
scarcely  more  intelligible  than  the  names  themselves. 

The  change  of  the  position  of  a  body  with  reference  to 
some  point  in  space  is  called  motion.  The  rate  at  which  a 
moving  body  changes  its  relative  position,  or  the  ratio  of 
the  distance  (/)  traversed  to  the  time  (  T)  is  called  its 
velocity  (u).  If  the  motion  is  not  uniform,  the  ratio  of  very 
small  intervals  of  space  and  time  must  be  taken ;  that  is, 
u  =  dl:  dr.  The  rate  at  which  a  moving  body  changes  its 
velocity  is  called  its  acceleration  (a) ;  that  is,  a  —  du:  dr. 

For  the  measurement  of  space,  time,  and  other  physical 
quantities,  a  great  variety  of  units  has  been  employed  at 
different  times  and  places  ;  but  in  scientific  considerations  a 
single  system  of  units,  known  as  the  absolute  or  centimeter- 
gram-second  (C.  G.  5.)  system,  has  now  been  almost  univer- 
sally adopted  as  the  standard  one.  Units  which  are  exact 
decimal  multiples  of  those  in  this  system  are,  however,  also 
commonly  employed.  The  unit  of  length  in  this  system 
is  the  centimeter  (cm.),  which  is  defined  to  be  one  one- 


FUNDAMENTAL    CONCEPTS  OF  PHYSICAL  SCIENCE.     U 

hundredth  part  of  the  distance  between  two  scratches  on 
a  standard  platinum-iridium  bar  kept  at  Paris,  when  the 
bar  is  at  the  temperature  of  melting  ice.  For  the  meas- 
urement of  surfaces  the  square  centimeter  (sqcm.),  and  for 
that  of  volumes  the  cubic  centimeter  (ccm.)  is  employed. 
Another  unit  for  the  measurement  of  volumes  is  also  in 
common  use,  especially  in  chemical  work ;  this  is  called  the 
liter  (£.),  and  is  by  definition  the  volume  of  1000  grams 
of  water  at  the  temperature  of  its  maximum  density :  this 
volume  is  very  nearly  equal  to  1000  ccm.  The  unit  of 
time  adopted  is  the  second,  or  -g-^Q-g-  part  of  a  mean  solar 
day,  which  is  the  average,  determined  throughout  the  year, 
of  the  intervals  of  time  elapsing  between  the  successive 
daily  transits  of  the  sun  over  the  meridian  at  any  place 
on  the  earth's  surface. 

6.  The  Concepts  of  Matter  and  Energy.  —  Experi- 
ence shows  that  different  portions  of  the  space  about  us, 
and  the  same  portion  of  space  at  different  times,  affect  our 
senses  differently,  giving  rise  to  distinct  mental  impres- 
sions, which  are  known  as  phenomena.  The  existence  of 
such  mental  impressions  leads  to  the  conclusion  that  there 
exist  in  our  surroundings  real  things  of  limited  extension 
which  give  rise  to  those  impressions ;  these  inferred  reali- 
ties are  called  objects,  bodies,  or  substances,  and  the  various 
powers  of  affecting  the  senses  which  they  possess  are  called 
their  properties.  The  choice  between  the  three  terms,  ob- 
ject, body,  and  substance,  depends  on  the  standpoint  from 
which  the  properties  are  considered.  The  thing  giving  rise 
to  the  impressions  is  called  an  object  when  it  is  desired 
to  denote  the  properties  of  a  definitely  bounded  portion  of 
space,  the  idea  of  a  definite  form  being  implied  ;  a  body 
when  it  is  desired  to  denote  the  properties  of  a  definite 
portion  of  matter ;  and  a  substance  when  the  properties  are 
considered  more  abstractly,  without  reference  either  to 
their  spacial  extension  or  their  association  with  any  defi- 
nite amount  of  matter.  Thus  we  speak  of  the  image  of  an 


12  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

object  in  a  mirror  or  of  the  shadow  cast  by  an  object,  for 
these  are  phenomena  in  which  the  form  of  the  thing  giving 
rise  to  the  impressions  comes  prominently  into  considera- 
tion ;  we  speak  of  the  attraction  of  two  bodies  for  one  an- 
other, or  the  work  done  by  a  moving  body,  or  of  the  laws 
of  falling  bodies,  for  these  are  phenomena  independent  of 
form  but  closely  connected  with  quantity  of  matter;  and, 
finally,  we  speak  of  the  solubility  or  the  melting-  or  boiling- 
point  of  a  substance,  for  these  properties  are  not  affected 
either  by  the  form  or  by  the  quantity  of  matter  considered. 
Many  properties,  to  be  sure,  like  color  or  hardness,  are 
common  to  objects,  bodies,  and  substances,  the  usage  then 
being  determined  by  the  connotation  in  the  directions  stated 
which  it  is  desired  to  give  to  the  term.  The  not  uncom- 
mon use,  especially  in  descriptive  chemistry,  of  the  word 
body  to  designate  a  chemical  substance,  is  inappropriate 
and  unnecessary.  The  properties  of  substances  when  so 
expressed  as  to  be  characteristic  of  them  and  independent 
of  the  quantity  of  matter  or  form  which  bodies  composed  of 
them  may  possess,  are  called  specific  properties ;  thus,  spe- 
cific gravity,  specific  volume,  crystalline  structure,  specific 
heat,  are  specific  properties  of  substances,  but  weight,  vol- 
ume, form,  and  heat  capacity  are  properties  of  bodies  rather 
than  of  substances. 

It  is  found  now  that  there  are  certain  properties,  specif- 
ically described  in  the  following  section,  which  are  possessed 
by  all  bodies,  and  which  in  a  definite  body  do  not  undergo 
a  change  in  magnitude  under  any  circumstances  whatever, 
no  matter  how  great  may  be  the  changes  in  its  other  proper- 
ties. These  facts  have  led  to  the  conception  of  the  exist- 
ence of  something  that  gives  rise  to  the  localization  of  the 
phenomena  from  which  are  inferred  the  existence  and  prop- 
erties of  bodies,  and  to  the  constancy  and  persistence  of 
certain  of  their  properties.  This  conceived  entity  is  called 
matter. 


FUNDAMENTAL    CONCEPTS   OF  PHYSICAL   SCIENCE.      13 

Most  of  the  properties  of  bodies,  however,  are  found 
to  undergo  great  changes,  some  of  them  entirely  disappear- 
ing, and  other  new  ones  taking  their  place.  This  fact  leads 
to  the  conception  that  there  exists  in  the  universe  some 
thing  or  things  other  than  matter  which,  by  association  with 
it,  gives  rise  to  the  changes  in  properties  which  bodies  ex- 
hibit, and  gives  to  them  the  power  of  producing  changes  in 
the  properties  of  other  bodies.  The  changes  produced  are 
very  varied  in  character ;  thus,  they  may  consist  of  changes 
in  the  relative  position,  the  state  of  motion,  the  temperature, 
the  volume,  the  form,  the  state  of  aggregation,  the  chemical 
composition,  or  in  almost  any  other  property,  of  the  bodies 
concerned.  The  powers  to  produce  such  changes,  moreover, 
arise  from  apparently  quite  distinct  characteristics  in  the 
bodies  possessing  them,  such  as  motion,  attraction,  difference 
of  pressure  or  temperature,  electrification,  and  chemical 
affinity.  It  is  found,  however,  that  such  powers  often  quan- 
titatively replace  one  another,  and  often  give  rise  to  identical 
effects.  The  changes  in  properties  and  the  power  to  pro- 
duce them  are  therefore  conceived  to  arise,  not  from  a 
number  of  distinct  entities,  but  from  a  single  one,  which 
is  capable,  however,  of  manifesting  itself  in  a  variety  of 
different  ways.  That  which  gives  rise  to  the  changes  in 
the  properties  of  bodies  and  to  the  power  to  produce  such 
changes  is  called  energy.  This  definition  is  obviously  only 
a  qualitative  one.  It  is  sufficient,  however,  to  enable  the 
presence  of  energy  to  be  recognized  ;  thus,  whenever  any 
change  in  the  properties  or  relations  of  bodies  occurs,  it 
shows  that  an  energy-change  is  taking  place,  and  whenever 
a  body  exhibits  the  power  of  producing  changes  in  other 
bodies,  it  is  to  be  inferred  that  it  possesses  energy. 

The  general  principles  relating  to  the  concept  of  matter, 
and  the  properties  of  bodies  most  closely  connected  with  it, 
will  be  discussed  in  Chapter  III ;  while  those  relating  to 
energy,  and  changes  in  the  properties  of  bodies  considered 
with  reference  to  it,  will  form  the  subject  of  Chapter  IV. 


CHAPTER    III. 

THE   GENERAL    PRINCIPLES    RELATING    TO    MATTER. 

7.  Matter,  its  Quantitative  Measurement,  and  the 
Law  of  its  Conservation.  —  Experience  shows  that  when 
a  moving  body  comes  in  contact  with  another  body  and 
thereby  parts  with  some  or  all  of  its  motion,  changes  are 
produced  in  the  condition  and  properties  of  both  bodies, 
for  example,  in  the  directions  and  rates  of  their  motion,  in 
their  forms,  or  in  their  temperatures.  A  moving  body  there- 
fore possesses  energy  in  virtue  of  its  motion :  such  energy 
is  called  kinetic  energy.  Now  it  is  found  that  the  kinetic 
energy  which  a  definite  body  moving  at  any  definite  velocity 
possesses  is  an  invariable  quantity  characteristic  of  the 
body,  and  entirely  independent  of  its  other  variable  prop- 
erties, such  as  its  volume,  form,  temperature,  or  state  of 
aggregation :  this  property  of  a  body  which  determines  the 
quantity  of  kinetic  energy  it  possesses  when  moving  at  a 
definite  velocity  is  called  its  capacity  for  kinetic  energy,  or 
its  mass.  Secondly,  experience  shows  that  any  body  located 
at  a  distance  from  another  body  tends  to  approach  it,  and 
does  in  fact  move  towards  it,  unless  prevented  by  some 
other  cause.  The  bodies  therefore  possess  energy  in  virtue 
of  their  relative  position  and  inherent  tendency  to  approach 
each  other :  such  energy  is  called  gravitation  energy.  Now, 
it  is  found  that  the  gravitation  energy  which  a  definite  body 
possesses,  when  placed  at  any  definite  distance  from  any 
definite  second  body,  is  an  invariable  characteristic  of  each 
of  the  two  bodies,  and  entirely  independent  of  the  other 
variable  properties  they  may  exhibit  :  this  property  of  a 
body,  in  virtue  of  which  it  possesses  under  the  stated 
conditions  a  definite  quantity  of  gravitation  energy,  may  be 


GENERAL  PRINCIPLES  RELATING    TO  MATTER.  15 

called  its  capacity  for  gravitation  energy.  This  quantity 
is  mathematically  defined  in  Chapter  IV. 

Thus  there  are  two  properties  of  a  body  which  are  per- 
sistent, inherent  ones,  not  detachable  from  the  body,  and 
therefore  differing  markedly  from  its  other  variable  proper- 
ties, which  are  only  temporarily  associated  with  it.  It  is 
therefore  natural  to  attribute  these  constant  properties  to 
the  existence  in  the  body  itself  of  something  which  gives 
rise  to  them,  an  assumption  which  simplifies  the  interpre- 
tation of  phenomena :  that  which  gives  rise  to  these  con- 
stant properties  is  called  matter. 

The  persistence  of  these  two  properties  is,  however,  by 
no  means  all  that  is  implied  in  the  concept  of  matter.  A 
further  inference  is  to  be  drawn  from  the  fact  that  every 
portion  of  space  which  is  capable  of  giving  rise  to  any  men- 
tal impression  whatever,  is  always  capable  of  giving  rise  to 
those  definite  impressions  from  which  are  inferred  the  prop- 
erties which  have  been  called  the  capacities  for  kinetic  and 
gravitation  energy.  This  inference  is,  that  matter  is  that 
which  gives  rise  to  the  localization  of  the  complex  of  prop- 
erties which  certain  portions  of  space  exhibit.  Even  though, 
on  the  one  hand,  it  must  be  admitted  that  'the  existence  of 
matter  is  inferred  only  from  the  various  energy  manifesta- 
tions which  bodies  exhibit,  it  must  be  acknowledged,  on  the 
other,  that  there  are  no  manifestations  of  energy  except 
those  which  are  associated  with  the  manifestations  of  it 
that  have  led  to  the  adoption  of  the  concept  of  matter :  in 
a  word,  the  two  assumed  entities,  matter  and  energy,  are 
indissolubly  connected  in  our  experience.  The  concept  of 
matter,  therefore,  would  still  remain  an  important  one,  even 
if  it  were  admitted,  as  some  scientists  have  contended,  that 
matter  can  be  fully  interpreted  in  terms  of  energy ;  for  it  is 
the  expression  of  certain  permanently  associated  relations 
which  are  involved  in  every  natural  phenomenon. 

The  two  properties  which  have  most  directly  led  to 
the  assumption  of  the  existence  of  matter  are,  as  has  been 


16  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

just  seen,  the  capacities  for  kinetic  and  gravitation  energy. 
These  two  quantities  are,  however,  in  the  case  of  definite 
bodies  strictly  proportional  to  each  other ;  for,  according  to 
Newton's  Law  of  Gravitation,  the  force  of  attraction  be- 
tween any  two  bodies  (or  the  degree  of  their  tendency  to 
approach  each  other)  is  directly  proportional  to  the  product 
of  their  masses,  and  inversely  proportional  to  the  square  of 
the  distance  between  them  ;  and  this  force  of  attraction, 
though  not  identical  with  the  invariable  property  of  bodies 
which  has  been  designated  their  capacity  for  gravitation 
energy,  is  nevertheless  proportional  to  it  in  the  case  of  dif- 
ferent bodies,  provided  the  force  of  their  attraction  be  meas- 
ured with  reference  to  a  definite  body  placed  at  a  definite 
distance  from  them. 

To  distinguish  these  two  properties,  the  brief  terms  mass 
and  weight  are  appropriately  employed.  These  may  be  de- 
fined as  follows,  some  standard  body  (like  the  kilogram-proto- 
type at  Paris)  being  first  adopted  as  a  unit  of  mass  and  weight : 
The  mass  of  any  body  is  the  ratio  of  the  kinetic  energy 
it  has,  to  that  which  the  unit-mass  has,  when  both  are  moving 
with  the  same  velocity.  The  weight  of  any  body  is  the 
ratio  of  the  gravitation  energy  it  has,  with  respect  to  the  earth, 
to  that  which  the  unit-weight  has,  when  both  are  located  at 
the  same  point.  Then  it  follows  from  Newton's  Law  that 
mass  and  weight  are  numerically  equal,  and  that  either  of 
them  may  be  taken  as  a  measure  of  quantity  of  matter  (m). 
Since  matter  is  an  abstract  concdpt  similarly  related 
both  to  kinetic  and  to  gravitation  energy,  and  since  it 
possesses  other  less  definite  connotations,  it  is  inappro- 
priate that  quantity  of  matter  should  be  designated  by 
either  of  the  concrete  terms  mass  or  weight.  Moreover, 
great  diversity  of  usage  exists  in  this  respect,  since  the 
term  mass  is  commonly  employed  by  physicists  on  account 
of  the  confusion  which  arises  in  physical  considerations 
from  the  use  of  the  term  weight  to  designate  both  a  force 
and  a  quantity  of  matter ;  while  the  term  weight  is  gener- 
ally employed  by  chemists,  since  it  is  a  direct  expression  of 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.          17 

the  property  actually  measured,  since  it  is  the  term  univer- 
sally employed  in  ordinary  life  and  in  commercial  transac- 
tions, and  since  the  identification  of  quantity  of  matter  with 
capacity  for  kinetic  energy,  mass,  is  at  least  equally  inappro- 
priate with  its  identification  with  capacity  for  gravitation 
energy,  or  with  weight.  In  order  to  avoid  this  impropriety 
of  using  one  word  to  express  two  distinct  ideas,  and  to  do 
away  with  the  diversity  of  usage  referred  to,  it  would  be 
necessary  to  introduce  at  least  one  new  term.  For  example, 
it  has  recently  been  suggested  to  use  the  word  mass  to  des- 
ignate quantity  of  matter  in  the  abstract,  kinergity  to  signify 
capacity  for  kinetic  energy,  and  weight  to  represent  the 
force  of  attraction  towards  the  earth.  Though  the  adoption 
of  some  new  third  term  would  be  conducive  to  much  greater 
clearness  in  dealing  with  the  fundamental  concepts  of  sci- 
ence, it  is  not  advisable  to  make  use  of  unusual  expressions 
in  an  elementary  treatise,  except  in  cases  where  their  em- 
ployment avoids  really  serious  confusion  in  the  subjects 
treated,  or  avoids  frequent  and  lengthy  circumlocutions. 
Throughout  this  work,  therefore,  the  term  weight  will  be 
commonly  employed,  in  accordance  with  the  usage  of  most 
leading  chemists,  to  designate  quantity  of  matter. 

The  unit-quantity  of  matter  employed  in  scientific  work 
is  called  the  gram,  and  is  defined  to  be  one  one-thousandth 
part  of  the  quantity  of  matter  in  a  standard  piece  of  plati- 
num-indium kept  at  Paris.  The  mass  and  weight  of  the 
gram  are  approximately  equal  to  those  of  a  cubic  centimeter 
of  pure  water  at  the  temperature  of  its  maximum  density. 
Recent  determinations  of  the  ratio  between  the  weight  of 
the  standard  gram  and  that  of  this  quantity  of  water  have 
given  as  the  most  probable  value  1:0.99996,  so  that  the 
difference  between  the  two  quantities  may  almost  always  be 
neglected. 

The  term  density  (D)  is  employed  to  designate  the  quan- 
tity of  matter  in  the  unit  of  volume ;  that  is,  D  =  m/*u. 
The  density  of  a  body  is  obviously  almost  exactly  equal, 


18  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

when  the  centimeter-gram-second  system  of  units  is  em- 
ployed, to  the  ratio  of  the  weight  of  any  definite  volume  of 
the  body  to  that  of  an  equal  volume  of  water  at  the  tem- 
perature of  its  maximum  density.  This  ratio  is  called  spe- 
cific gtavity.  The  reciprocal  of  the  specific  gravity  or  den- 
sity of  a  body,  or  the  volume  occupied  by  one  gram  of  it,  is 
called  its  specific  volume  (2) ;  that  is,  v  =  v/m. 

When  two  or  more  different  substances  are  uniformly 
distributed  throughout  the  same  space,  it  is  customary  to 
employ  the  term  concentration  (c)  to  designate  the  weight 
of  any  one  of  them  in  the  unit  of  volume. 

It  has  been  seen  that,  however  radical  may  be  the 
changes  produced  in  the  other  properties  of  a  body,  its 
mass  and  its  weight  determined  under  the  same  conditions 
remain  unchanged,  a  fact  which  is  commonly  expressed, 
adopting  the  concept  of  matter,  by  the  statement  that  mat- 
ter is  not  created  nor  destroyed  by  any  process  whatever. 
This  principle  is  known  as  the  Law  of  the  Conservation  of 
Matter. 

This  fundamental  law  has  been  inductively  derived 
from  a  very  large  number  of  experiments,  including  many 
chemical  analyses  and  syntheses.  Recent  experiments,  in 
which  chemical  reactions  were  caused  to  take  place  in  closed 
vessels  whose  weights  were  determined  before  and  after  the 
change,  have  shown  that  the  variation  in  the  weight  of  the 
reacting  bodies  certainly  does  not  exceed  one  one-millionth 
part.  Thus  in  one  experiment,  86.75  grams  of  silver  sul- 
phate and  238.25  grams  of  water  were  placed  in  one  arm  of 
a  n  -shaped  glass  tube  and  200  grams  of  crystallized  ferrous 
sulphate  and  125  grams  of  dilute  sulphuric  acid  in  the  other 
arm,  and  the  tube  was  hermetically  sealed  and  weighed. 
The  tube  was  then  inverted,  whereby  the  two  liquids  be- 
came mixed  and  a  reaction  took  place,  resulting  in  the  pro- 
duction of  metallic  silver  and  ferric  sulphate.  The  tube 
was  then  weighed  again,  and  was  found  to  have  an  apparent 
weight  of  0.00013  gram  less  than  that  at  first,  correspond- 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.  19 


ing  to  a  decrease  of  only  ^^-1-^^  part  of  the  weight  of  the 
reacting  ferrous  and  silver  sulphates.  As  this  small  appar- 
ent decrease  may  well  have  been  due  to  error  in  weighing, 
no  significance  is  to  be  attributed  to  it.  But  it  is  to  be 
noted  that  this  experiment  only  proves  that  the  Law  of  the 
Conservation  of  Matter  is  not  inaccurate  in  this  case  by  a 
greater  amount  than  that  mentioned.  This  illustrates,  more- 
over, the  general  principle  that,  since  all  measurements  are 
subject  to  a  greater  or  less  experimental  error,  it  is  not  pos- 
sible to  prove  the  absolute  exactness  of  any  law.  It  is, 
therefore,  always  important  to  state  within  what  limits  the 
accuracy  of  a  law  has  been  tested. 

8.  The  States  of  Aggregation  and  Other  Physical 
States  of  Matter.  —  When  to  a  homogeneous  solid  sub- 
stance, like  a  mass  of  ice,  kept  under  a  definite  pressure, 
for  example  that  of  the  atmosphere,  heat  is  continuously 
imparted,  the  following  phenomena  are  generally  observed  : 
The  substance  first  exhibits  a  steady  rise  in  temperature 
attended  by  a  slight  increase  in  volume.  When  a  definite 
temperature  has  been  reached,  no  further  rise  of  tempera- 
ture occurs  for  a  time,  but  the  substance  ceases  to  be  homo- 
geneous, and  becomes  resolved  into  two  parts,  one  of  which, 
in  the  so-called  liquid  state,  possesses  very  different  charac- 
teristics from  those  of  the  other  unchanged  solid  part. 
This  definite  temperature,  which  is  that  at  which  the  solid 
and  liquid  states  of  the  substance  coexist  in  equilibrium 
with  each  other,  is  called  its  'melting-point  ;  it  varies  slightly 
with  the  pressure  to  which  the  substance  is  subjected. 
When  the  solid  part  has  disappeared  and  the  substance  has 
become  a  homogeneous  liquid,  a  continuous  rise  of  tem- 
perature again  takes  place,  attended  generally  by  a  small 
increase  in  volume,  until  another  definite  temperature  is 
reached,  at  which  the  substance  again  loses  its  homogeneity, 
going  over  first  partly  and  then  completely  into  the  so-called 
gaseous  state,  in  which  it  exhibits  an  entirely  new  set  of 
properties.  This  temperature,  which  is  that  at  which  under 


20  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

any  definite  pressure  the  liquid  and  gaseous  states  of  the 
substance  coexist  in  equilibrium  with  each  other,  is  called 
its  boiling-point ;  it  varies  greatly  with  the  pressure  upon 
the  substance.  Finally,  after  the  substance  has  gone  over 
completely  into  the  gaseous  state,  its  temperature  again 
begins  to  rise,  and  it  continues  to  do  so  indefinitely  as  long 
as  heat  is  imparted ;  this  rise  of  temperature  being  attended 
by  a  relatively  large,  continuous  increase  in  volume.  If,  on 
the  other  hand,  heat  is  withdrawn  from  the  gas,  the  sub- 
stance undergoes  the  same  changes  in  the  reverse  order.  It 
should  be  added,  however,  that  this  behavior  is  exhibited  in 
all  its  details  only  by  pure  chemical  substances  (§9),  and  that 
some  of  these,  when  under  certain  pressures,  pass  directly 
from  the  solid  to  the  gaseous  state,  and  vice  versa,  with- 
out assuming  an  intermediate  liquid  state. 

The  passage  from  the  liquid  or  solid  to  the  gaseous 
state  can  always  be  brought  about  by  decrease  of  pressure, 
as  well  as  by  increase  of  temperature.  Thus,  if  at  a  defi- 
nite temperature  the  pressure  on  a  liquid  or  solid  substance 
is  steadily  reduced,  it  retains  its  state,  increasing  slightly  in 
volume,  until  a  definite  pressure  is  reached ;  it  then  begins 
to  go  over  into  the  gaseous  state,  and  the  pressure  can  not 
be  further  reduced  until  the  whole  substance  has  become  a 
homogeneous  gas.  Then  the  pressure  can  be  diminished 
and  the  volume  of  the  substance  increased  indefinitely  with- 
out its  becoming  again  heterogeneous.  The  pressure  at 
which  at  any  definite  temperature  the  liquid  or  solid  state 
and  the  gaseous  state  of  the  substance  coexist  in  equilibrium 
with  each  other  is  called  its  vapor-pressure.  A  clear  con- 
ception of  this  property  is  so  important  that  an  experiment 
illustrating  it  will  be  here  described.  A  tube  closed  at  one 
end,  100  cm.  in  length,  is  filled  with  mercury,  and  is  inverted 
over  a  deep  trough  also  containing  mercury.  The  mercury 
column  in  the  tube  then  stands  at  a  definite  height,  say  76 
cm.,  above  the  free  surface  of  the  mercury  in  the  trough,  its 
weight  being  supported  by  the  pressure  of  the  atmosphere 


GENERAL  PRINCIPLES  RELATING    TO  MATTER.          21 

upon  that  surface.  The  tube  is  kept  at  a  constant  tempera- 
ture, say  20°.  At  the  bottom  of  the  tube  a  very  little  liquid 
ether  is  introduced,  and  this  rises  into  the  vacuum  above  the 
mercury,  and  completely  vaporizes.  The  mercury  column 
then  falls,  owing  to  the  pressure  of  the  ether-vapor  upon  its 
upper  surface.  Small  additional  quantities  of  ether  are 
successively  introduced,  and  the  column  continues  to  fall 
until  such  a  quantity  of  ether  is  present  that  it  no  longer 
completely  vaporizes,  but  remains  in  part  in  the  liquid  state. 
Further  additions  of  ether  then  have  no  further  influence  on 
the  height  of  the  mercury  column,  which  remains  constant 
at  33  cm.,  the  difference  (43  cm.)  from  the  original  height 
(76  cm.)  representing  the  vapor-pressure  of  ether  at  20°. 
If  the  tube  is  now  pushed  down  into  the  mercury  trough,  it 
becomes  more  nearly  filled  with  mercury,  the  space  occupied 
by  the  ether-vapor  is  reduced,  and  some  of  it  liquefies,  but 
the  height  of  the  top  of  the  mercury  column  above  the  free 
surface  of  the  mercury  remains  unchanged  at  33  cm.,  since 
the  pressure  exerted  by  the  ether,  its  vapor-pressure,  remains 
unchanged.  If  finally  the  tube  is  pushed  down  so  far  that 
the  length  of  it  above  the  free  mercury  surface  is  less  than 
33  cm.,  all  the  ether  liquefies  ;  for  the  pressure  exerted  upon 
it,  which  is  always  that  of  the  atmosphere  diminished  by 
that  corresponding  to  the  weight  of  the  column  of  mercury> 
is  now  greater  than  the  vapor-pressure  (43  cm.)  of  ether  at 
20°.  —  The  vapor-pressure  of  a  liquid  or  solid  increases 
rapidly  with  rise  of  temperature ;  thus,  expressed  in  centi- 
meters of  mercury,  that  of  ice  is  0.19  at —  10°,  and  0.46  at 
o°,  and  that  of  water  is  0.46  at  o°,  2.35  at  25°,  9.20  at  50°, 
28.9  at  75°,  76.0  at  100°,  and  174.4  at  I25°-  When  a  liquid 
has  acquired,  through  an  increase  in  its  temperature  a  vapor- 
pressure  which  is  greater  by  an  indefinitely  small  amount 
than  the  external  pressure  to  which  it  is  subjected,  it  will 
evidently  drive  back  this  pressure  and  go  over  into  the 
gaseous  state  ;  in  other  words,  it  will  boil.  The  boiling-point 
of  a  liquid  is  therefore  that  temperature  at  which  its  vapor- 


22  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

pressure  is  equal  to  the  external  pressure  under  which  it 
stands.  Thus,  water  boils  at  100°  under  the  normal  atmos- 
pheric pressure  of  76  cm.,  for  its  vapor-pressure  is  76  cm.  at 
that  temperature ;  and  from  the  vapor-pressure  values  just 
given  it  is  evident  that  it  would  boil  at  50°  if  the  external 
pressure  were  reduced  to  9.2  cm.,  and  at  o°  if  it  were  re- 
duced to  0.46  cm. 

A  substance  can  be  made  to  pass  also  from  the  solid 
to  the  liquid,  or  from  the  liquid  to  the  solid  state,  by  varia- 
tions of  pressure ;  but  if  it  is  not  near  its  melting-point, 
the  changes  of  pressure  required  are  enormous.  Moreover, 
increase  of  pressure  tends  to  cause  some  substances  to 
assume  the  solid  state,  and  others,  the  liquid  state ;  the 
direction  of  its  influence  is  therefore  not  uniform. 

These  three  different  states  into  which  matter  can  be 
brought  by  varying  the  temperature  or  pressure,  and  in  one 
or  more  of  which  every  substance  exists,  are  called  its  states 
of  aggregation.  This  name  has  been  adopted  because  it  is 
commonly  assumed  that  in  the  different  states  the  particles 
composing  the  substance  are  differently  aggregated  to  form 
the  mass. 

The  characteristics  of  substances  in  the  three  states 
may  be  next  considered.  The  behavior  under  diminution 
of  pressure,  which  has  been  just  described,  furnishes  the 
best  basis  for  a  definition  of  the  gaseous  state  of  aggrega- 
tion. That  is,  a  substance  is  called  a  gas  if  it  increases  in 
volume  indefinitely  without  losing  its  homogeneity  when  the 
pressure  upon  it  is  diminished  and  its  temperature  is  kept 
constant.  Under  the  usual  conditions  of  pressure  and  tem- 
perature gases  differ  .from  liquids  and  solids  also  in  the  fol- 
lowing respects  :  the  volumes  of  gases  vary  much  more  with 
variations  of  pressure  and  temperature  4than  do  those  of 
liquids  or  solids ;  and  the  densities  of  the  former  are  very 
much  less  than  those  of  the  latter.  Thus,  if  the  pressure 
upon  a  quantity  of  air  at  o°  under  the  pressure  of  the 
atmosphere  is  doubled,  its  volume  is  reduced  50  per  cent., 


GENERAL  PRINCIPLES  RELATING    TO  MATTER.          23 

while  the  same  change  of  pressure  causes  a  decrease  in  the 
volume  of  water  at  o°  of  only  0.005  Per  cent-  J  ^  a*r  at  °°  is 
heated  to  100°,  keeping  the  pressure  constant,  it  increases 
in  volume  by  37  per  cent.,  while  water  increases  in  volume 
by  4.3  per  cent,  between  the  same  temperatures.  The 
density  of  air  at  o°  under  the  pressure  of  the  atmosphere 
is  0.0013,  while  that  of  water  under  the  same  conditions  is 
0.9999,  or  770  times  as  great,  and  that  of  ice  is  0.917,  or 
about  700  times  as  great.  Highly  compressed  gases,  how- 
ever, do  not  differ  greatly  from  liquids  in  these  respects ; 
in  fact,  by  suitable  variations  of  temperature  and  pressure 
the  two  states  can  be  made  to  pass  over  continuously  into 
each  other. 

It  is  found  that  gases  can  always  be  converted  into 
liquids  by  sufficiently  reducing  their  temperature,  but  that, 
when  the  temperature  exceeds  a  definite  value,  called  the 
critical  temperature,  which  varies  with  the  nature  of  the 
gas,  they  cannot  be  so  converted  by  increasing  the  pressure 
upon  them  while  keeping  the  temperature  constant.  Thus, 
air  at  the  ordinary  temperature  does  not  become  liquid  even 
when  subjected  to  a  pressure  of  3600  atmospheres.  This 
difference  in  behavior  has  led  to  the  employment  of  differ- 
ent terms  to  distinguish  the  two  classes  of  gases.  Those 
which  can  be  condensed  to  liquids  by  pressure  alone  are 
called  vapors,  while  those  which  can  not  be  so  condensed 
are  called  gases,  the  last  term  being  used  in  this  narrower 
sense  when  it  is  wished  to  contrast  these  two  kinds  of 
behavior.  The  distinction  is,  however,  one  of  compara- 
tively small  importance;  for  the  properties  of  vapors  and 
gases  are  essentially  the  same. 

Liquids  and  solids,  even  under  the  ordinary  conditions, 
do  not  show  any  uniform  differences  in  behavior  when  the 
pressure  upon  them  is  varied,  nor  any  regular  differences 
in  their  degrees  of  compressibility  or  expansibility  by  heat, 
or  in  their  densities.  In  a  word,  their  w/ww^-relations  are 
substantially  the  same.  Their  distinguishing  characteristic 


24  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

is  their  difference  in  behavior  with  respect  to  changes  in 
form.      A   substance  is   called  a  liquid  when   its  form   is 
•greatly  affected  even  by  very  slight  external  forces.    A  sub- 
stance is  called  a  solid  when  its  form  is  appreciably  changed 
only  by  the  application  of  considerable  external  forces.    Since 
the  difference  is  one  of   degree,  substances,  like   soft  tar, 
which  are  in  an    intermediate   state  that   can    scarcely  be 
designated   either  solid  or  liquid,  are  sometimes  met  with. 
Under   ordinary    conditions    of   pressure   and   temperature, 
however,   it  is   relatively  rare  to  find   pure   chemical   sub- 
stances which  do  not  exhibit  a  marked  difference  with  re- 
spect to  stability  of  form  in  the  two  states  of  aggregation. 
Solid   substances   are,  moreover,  sub-divided  into  two 
groups,  designated  crystalline   and  amorphous   substances, 
which  differ  markedly  in  behavior  and  properties.     One  of 
the  most  important  of  these  differences  is  that  shown  when 
the  substances  are  sufficiently  heated  to  cause  them  to  pass 
into   the   liquid   state.      A   crystalline    substance,  like  ice, 
when  heated,  retains  the  stability  of  form  characteristic  of 
solid  substances  until  a  definite  temperature  is  reached,  at 
which  it  becomes  liquid  and  undergoes  at  once  a  decided 
change  in  its   properties ;   that  is,  a  crystalline    substance 
possesses   a   definite    melting-point.      An    amorphous   sub- 
stance, on  the  other  hand,  like  glass,  when  heated,  gradually 
loses  its  stability  of  form  and  passes  over  into  the  liquid 
state  continuously,  that  is,  without  undergoing  at  any  tem- 
perature a  sudden  change    in    properties.     An    amorphous 
substance    has    no    definite    melting-point,    but    gradually 
softens  as  the  temperature  rises.     The  two  groups  of  sub- 
stances when  in  the  form  of  homogeneous  masses  differ  also 
in  the  following  important  respect.     It  is  always  true  that 
some  of  the  properties  of   homogeneous  crystalline  bodies 
{the  so-called  crystals)  are  different  in  different  directions 
through  the  crystals,  but  all  the  properties  of  homogeneous 
amorphous  bodies  are  the  same  in  all  directions.     For  ex- 
ample, nearly  a  third  more  tensile  force  has  to  be  applied  in 


GENERAL  PRINCIPLES  RELATING    TO  MATTER.          25 

order  to  lengthen  by  a  definite  amount  under  corresponding 
conditions  a  piece  of  rock-salt  in  the  direction  of  one  of  its 
so-called  crystallographic  axes  than  in  a  direction  making  an 
angle  of  45°  with  these ;  glass,  on  the  contrary,  is  extended 
with  equal  readiness  in  all  directions.  But,  since  extremely 
small  crystalline  particles  may  be  irregularly  aggregated  to 
form  an  apparently  homogeneous  body,  which  would  not 
exhibit  difference  of  properties  in  different  directions,  this 
characteristic  is  not,  from  a  chemical  standpoint,  as  satis- 
factory a  criterion  of  crystalline  substances  as  their  defi- 
niteness  of  melting-point. 

Solid  and  liquid  substances  are  also  naturally  sub- 
divided, on  the  basis  of  certain  other  differences  in  proper- 
ties, into  two  groups,  known  as  metallic  and  non-metallic 
substances.  Although  here  again  the  differences  are  those 
of  degree  rather  than  of  kind,  and  although  a  few  sub- 
stances occupy  an  intermediate  position,  yet  in  the  great 
majority  of  cases  the  differences  in  properties  are  so  marked 
as  to  make  the  distinction  one  of  some  importance.  A 
metallic  substance  is  a  substance  which  transmits  light  only 
through  extremely  thin  layers  and  conducts  electricity 
readily  without  undergoing  any  other  changes  than  those 
produced  by  a  rise  in  its  temperature.  A  non-metallic  sub- 
stance is  one  which  transmits  light  through  much  greater 
thicknesses  and  either  does  not  conduct  electricity  with  any 
degree  of  readiness,  or  if  it  does  conduct  it,  undergoes  an 
attendant  change  in  chemical  composition.  Thus  a  centi- 
meter-cube of  copper  allows  electricity  to  pass  through  it 
freely  and  undergoes  thereby  only  a  rise  of  temperature ; 
on  the  other  hand,  one  of  rock-salt  at  the  ordinary  tempera- 
ture allows  scarcely  any  electricity  to  pass  through  it ;  and 
though  it  can  be  made  to  conduct  fairly  readily  by  suffi- 
ciently raising  its  temperature  or  dissolving  it  in  water,  the 
passage  of  electricity  through  it  is  then  attended  by  a 
partial  decomposition  into  chlorine  and  sodium,  or  into 
chlorine,  sodium  hydroxide,  and  hydrogen. 


26  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

Still  another  physical  state  of  importance  is  that  in 
which  two  or  more  different  substances  are  so  intimately 
mixed  with  one  another  that  the  masses  of  one  or  all  of 
them  become  subdivided  into  minute  particles  which  are 
uniformly  distributed  throughout  the  whole  volume.  Such 
intimate  mixtures  are  designated  by  different  names  accord- 
ing to  the  degree  of  heterogeneity  which  they  exhibit.  If 
the  particles  of  one  of  the  components  of  the  mixture  are 
large  enough  to  be  seen  by  the  eye  with  the  aid  of  the  best 
microscope,  the  mixture  is  called  a  suspension  when  the 
visible  particles  are  those  of  a  solid  substance,  and  an  emul- 
sion when  the  visible  particles  are  those  of  a  liquid  sub- 
stance. When  the  particles  are  of  such  a  size  that  the 
mixture  exhibits  no  indication  of  heterogeneity  under  the 
best  microscope,  but  nevertheless  affects  a  beam  of  ordinary 
light  thrown  into  it  in  the  same  way  as  fine  suspensions 
affect  it,  namely,  causes  the  beam  to  become  scattered  and 
the  light  to  become  polarized,  the  mixture  is  called  a  col- 
loidal or  pseudo-solution.  Finally,  if  the  particles  are  so  small 
that  no  indication  of  heterogeneity  can  be  detected  by  any 
physical  instrument  (even  by  a  microscope  or  polariscope), 
the  mixture  is  called  a  solution.  Thus,  the  intimate  mixture 
formed  by  agitating  finely  divided  clay  with  water  is  a  sus- 
pension, that  made  by  violently  shaking  a  little  olive  oil 
with  water  is  an  emulsion,  that  prepared  by  passing  hydro- 
gen sulphide  into  a  boiling  solution  of  arsenious  oxide  is  a 
colloidal  solution,  and  that  made  by  adding  salt  to  water  is 
an  ordinary  solution.  Such  mixtures  may  be  gaseous,  liquid, 
or  solid ;  but  liquid  mixtures  are  the  most  important. 

The  various  kinds  of  mixtures  which  have  just  been 
described  can  in  many  cases  be  separated  into  their  compo- 
nents by  enclosing  them  within  suitable  porous  walls  or 
membranes  and  exerting  a  pressure  upon  them.  Walls 
which  permit  the  passage  of  one  component  of  a  mixture, 
but  prevent  entirely  that  of  the  other  component,  are 
called  semipermeable  walls.  Different  kinds  of  semiper- 


GENERAL  PRINCIPLES  RELATING    TO  MATTER.          27 

meable  walls  must,  however,  be  employed  for  the  separa- 
tion of  the  components  of  suspensions,  colloidal  solutions, 
and  ordinary  solutions,  and  this  fact  furnishes  another  basis 
for  the  differentiation  of  these  three  kinds  of  mixtures. 
Thus,  filter  paper  will  usually  separate  solid  particles  from 
a  liquid  in  which  they  are  suspended ;  but  parchment  or 
animal  membranes  must  be  employed  for  the  separation  of 
the  components  of  colloidal  solutions ;  and  specially  pre- 
pared semipermeable  walls,  such  as  are  produced  by  the 
deposition  of  precipitates  like  copper  ferrocyanide  in  the 
pores  of  unglazed  porcelain,  must  be  used  for  resolving  ordi- 
nary solutions,  like  those  of  salt  or  sugar  in  water,  into 
their  constituents. 

When  the  concentration  of  one  component  of  a  solu- 
tion of  two  substances  greatly  exceeds  that  of  the  other 
component,  the  solution  is  said  to  be  dilute  with  reference 
to  the  latter.  In  such  a  case,  the  component  present  in 
large  amount  is  called  the  solvent,  the  one  present  in  small 
amount,  the  solute  or  dissolved  substance.  If  the  concentra- 
tion of  neither  component  greatly  exceeds  that  of  the  other, 
the  solution  is  concentrated  with  respect  to  either  component, 
and  either  may  be  designated  the  solvent  or  solute  according 
to  the  relations  under  consideration.  In  the  case  of  col- 
loidal solutions,  the  substance  which  possesses  the  larger 
particles  is  called  the  colloid,  and  the  other  component  is 
called  the  solvent. 

Another  term,  which  expresses  from  a  different  point 
of  view  the  degree  of  heterogeneity  of  bodies,  and  which  is 
much  employed  in  considerations  relating  to  their  equilib- 
rium, will  be  here  defined.  Heterogeneous  bodies  in  a 
state  of  equilibrium  are  considered  to  consist  of  a  number 
of  parts  which  are  differentiated  by  the  possession  of  dis- 
tinct sets  of  properties,  and  by  their  separation  from  one 
another  by  sharp  physical  boundaries,  each  part  being  char- 
acterized by  uniformity  in  properties  throughout  its  extent. 
These  physically  distinct  parts  are  called  phases.  Thus,  ice 


28  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

and  water,  water  and  water-vapor,  or  salt  and  its  saturated 
aqueous  solution,  when  existing  together,  constitute  two 
distinct  phases.  A  body  consisting  of  ice,  a  solid  salt,  an 
aqueous  solution  of  the  salt,  and  water-vapor  consists  of 
four  phases,  two  of  which  are  solid,  one  liquid,  and  one 
gaseous ;  a  body  composed  of  liquid  benzene  and  water 
consists  of  two  phases,  both  of  which  are  liquid ;  one  com- 
posed of  hydrogen,  oxygen,  and  liquid  water,  consists  also 
of  two  phases,  one  of  which  is  gaseous  and  the  other  liquid  ; 
but  one  composed  of  hydrogen,  oxygen,  and  water-vapor, 
consists  of  only  a  single  (gaseous)  phase.  It  is  to  be  noted 
that  the  terms  homogeneous,  heterogeneous,  and  phase,  are 
always  used  in  a  physical  sense,  entirely  without  reference 
to  the  number  of  chemical  substances  that  may  be  present. 
9.  Chemical  Substances  and  Mixtures.  —  Most,  but 
not  all,  of  the  numerous  substances  met  with  in  nature,  or 
produced  artificially,  can  be  resolved  by  certain  kinds  of 
processes  which  have  been  found  suitable  for  the  differen- 
tiation of  substances  in  the  manner  to  be  now  considered, 
into  parts  or  fractions  possessing  properties  different  from 
those  of  the  original  substances.  A  substance  which  can 
not  be  resolved  by  such  processes  into  component  sub- 
stances with  different  properties  is  called  a  chemical  sub- 
stance, or  often  simply  a  pure  substance.  All  substances 
which  can  be  so  resolved  are  called  mixtures ;  for  they  can 
be  produced  by  mixing  pure  chemical  substances.  In  ac- 
cordance with  these  definitions,  the  method  to  be  employed 
for  distinguishing  chemical  substances  from  mixtures  con- 
sists in  submitting  the  material  to  various  processes  of 
fractionation  —  that  is,  to  operations  which  result  in  divid- 
ing it  into  separate  parts  or  fractions,  and  in  then  determin- 
ing whether  or  not  the  properties  of  the  different  fractions 
are  identical.  If  identical,  the  material  is  probably  a  pure 
substance :  it  may,  however,  be  a  mixture  which  the  proc- 
esses of  fractionation  employed  have  failed  to  separate  into 
the  substances  composing  it.  On  account  of  this  last  pos- 


GENERAL  PRINCIPLES  RELATING    TO   MATTER.          29 

sibility,  as  many  distinct  methods  of  fractionation  as  possi- 
ble must  be  resorted  to,  and  only  if  the  fractions  obtained 
by  all  methods  are  identical  in  properties,  is  the  material  to 
be  regarded  as  a  pure  chemical  substance.  If  the  fractions 
are  not  identical,  the  material  is  probably  a  mixture :  it  may, 
however,  be  a  pure  substance  which  the  processes  of  frac- 
tionation employed  have  converted  in  part  into  a  different 
chemical  substance.  In  order  to  avoid  an  erroneous  conclu- 
sion from  this  source,  only  those  processes  of  fractionation 
should  be  employed  which  have  been  shown  by  experience 
not  to  give  rise  as  a  rule  to  chemical  transformations. 

A  more  definite  statement  than  that  just  made,  of  the 
character  of  the  processes  suitable  for  distinguishing  chem- 
ical substances  from  mixtures  can  not  be  formulated,  since 
the  processes  that  must  be  applied  vary  greatly  with  the 
nature  of  the  material.  Some  specific  illustrations  of  the 
processes  of  fractionation  commonly  employed  in  the  labora- 
tory for  determining  the  chemical  purity  of  substances  may, 
however,  be  presented.  If  the  material  is  solid,  it  may  be 
suspended  in  liquids  of  varying  specific  gravity  in  which  it 
is  insoluble,  it  may  be  partially  melted,  it  may  be  treated 
with  solvents  in  such  quantity  that  only  a  portion  is  dis- 
solved, or  it  may  be  completely  dissolved  and  caused  to  crys- 
tallize from  the  solvent  in  separate  portions  by  cooling  or 
evaporation.  If  the  material  is  liquid,  it  may  be  fraction- 
ated by  partial  freezing  or  partial  volatilization,  or  by  dialy- 
sis. If  the  material  is  gaseous,  it  may  be  resolved  into  frac- 
tions by  causing  it  partially  to  diffuse  through  porous  walls, 
or  by  treating  it  with  liqfuid  solvents  in  insufficient  amount 
to  dissolve  the  whole  of  the  gas  and  expelling  the  dis- 
solved part  from  the  solvent  by  decrease  of  pressure  or 
rise  of  temperature.  On  the  other  hand,  processes  involv- 
ing the  exposure  of  the  substance  to  a  high  temperature 
(like  the  distillation  of  a  difficultly  volatile  substance),  the 
subjection  of  it  to  the  action  of  an  electric  current  or  dis- 
charge, or  the  treatment  of  it  with  solvents  of  great  chemi- 


30  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

cal  activity  (like  acids  or  bases),  are  not  suitable  ones  for 
attaining  the  end  in  question. 

The  properties  employed  for  determining  the  identity 
of  the  various  fractions  are  any  quantitative  ones  which 
can  be  conveniently  measured,  such  as  chemical  composi- 
tion, melting-point,  boiling-point,  specific  gravity,  crystalline 
form,  spectrum,  etc. 

Although  the  above  described  means  of  distinguishing 
mixtures  and  chemical  substances  is  the  one  which  is 
usually  first  employed  by  the  investigator  and  is  one  which 
in  almost  all  cases  leads  to  a  correct  conclusion,  it  is  never- 
theless somewhat  indefinite  and  not  always  decisive ;  for 
some  substances  which  in  other  respects  behave  like  pure 
chemical  compounds,  are  so  unstable  that  even  the  ordinary 
processes  of  fractionation  cause  their  decomposition ;  and, 
on  the  other  hand,  some  mixtures  consist  of  substances  so 
similar  in  properties  that  such  processes  fail  to  effect  sepa- 
ration. Other  characteristic  differences  which  have  been 
found  to  exist  between  mixtures  and  chemical  substances  of 
known  character  must  therefore  be  considered.  One  such 
characteristic  difference  of  importance  is  that  the  specific 
properties  of  mixtures  are  at  least  approximately  additive, 
that  is,  approximately  the  average  of  the  properties  of  the 
substances  into  which  they  can  be  separated,  while  many 
of  the  properties  of  pure  chemical  substances  seem  to 
bear  scarcely  any  relation  to  those  of  their  decomposition- 
products.  A  chemical  substance  may,  therefore,  also  be 
defined  as  a  substance  many  of  whose  properties  are  not 
even  approximately  the  average  of  the  properties  of  the 
substances  into  which  it  can  be  resolved  by  any  method 
whatever. 

For  example,  water  can  be  decomposed  by  the  electric 
current  into  hydrogen  and  oxygen;  but  its  properties  are 
not  in  the  least  similar  to  those  of  these  two  gases.  On  the 
other  hand,  a  mixture  of  hydrogen  and  oxygen,  though  sepa- 
rable into  its  component  substances  only  imperfectly  and 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.          31 

with  difficulty,  has  specific  properties  which  are  the  aver- 
age of  those  of  the  two  separate  gases.  If  sugar  is  sub- 
mitted to  distillation  it  is  resolved  into  distinct  parts,  namely, 
into  a  black  solid,  a  brown  liquid,  and  a  colorless  gas ;  but 
it  would  be  obviously  incorrect  to  conclude  from  this  fact 
that  the  sugar  was  a  mixture,  for  most  of  its  properties  are 
not  in  the  least  related  to  those  of  the  fractions  into  which 
it  has  been  resolved.  Similarly,  if  metallic  sodium  is 
treated  with  an  amount  of  water  insufficient  to  dissolve  it, 
a  gas  and  a  white  solid  result,  whose  properties  differ  utterly 
from  those  of  the  original  metal ;  it  is,  therefore,  clear  that 
the  process  has  resulted  in  chemical  decomposition  and  is 
not  a  suitable  one  for  determining  the  purity  of  the  sodium* 
The  last  two  examples  are  of  an  extreme  character,  since 
in  each  case  the  decomposition-products  differ  so  markedly 
from  the  original  substance,  and  since  decomposition  of  a 
large  part  or  of  all  of  the  original  substance  takes  place. 
In  some  cases  a  careful  investigation  is  necessary,  in  order 
to  distinguish  the  effect  of  a  slight  decomposition  from 
that  of  an  impurity  originally  present. 

In  distinguishing  mixtures  from  pure  chemical  sub- 
stances, it  is  best  to  apply  in  any  doubtful  case  both  of  the 
criteria  which  have  been  described ;  that  is,  first  to  deter- 
mine whether  the  substance  can  be  resolved  into  fractions 
with  different  properties  by  the  commonly  employed  proc- 
esses of  fractionation,  and  then  to  determine  whether  the 
properties  of  the  original  substance  are  additive  with  respect 
to  those  of  these  fractions :  if  not  additive,  the  method  of 
fractionation  employed  must  be  rejected  as  unsuitable  for 
attaining  the  end  in  view. 

A  third,  important,  though  not  sufficient,  criterion  of 
a  pure  chemical  substance  is  furnished  by  its  conformity 
with  the  laws  relating  to  elementary  composition,  which 
have  been  proved  to  hold  true  in  the  case  of  all  substances 
determined  to  be  pure  by  the  two  methods  already  described. 
These  laws  and  their  value  as  a  means  of  distinguishing 
chemical  substances  and  mixtures  are  considered  below. 


32  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

Since  the  properties  of  all  substances  vary  to  a  greater 
or  less  extent  with  the  temperature  and  pressure,  and  since 
the  identity  of  two  substances  can  be  established  only  by 
comparing  their  properties  under  the  same  conditions,  the 
question  arises,  whether  a  definite  portion  of  matter  is  to 
be  regarded  as  the  same  chemical  substance,  when  it  exists 
in  different  physical  states.  That  pure  chemical  substances 
are  sometimes  transformed  into  other  chemical  substances 
merely  by  variations  of  temperature  and  pressure  is  proved 
by  the  fact  that  they  sometimes  become  mixtures  ;  thus,  the 
vapor  produced  when  solid  ammonium  chloride  is  heated  can 
be  shown  by  diffusion  experiments  to  be  a  mixture  of  ammo- 
nia and  hydrochloric  acid  gases.  Even  when  a  mixture  does 
not  result,  it  is  evidently  somewhat  arbitrary  to  attribute  a 
radical  change  in  the  properties  of  a  substance  to  a  change 
in  its  physical  state  rather  than  to  a  complete  transforma- 
tion of  it  into  another  chemical  substance.  Nevertheless, 
the  practice  which  is  commonly  followed  is  to  regard  pure 
substances  which  pass  over  into  one  another  completely  and 
reversibly  by  variations  of  temperature  or  pressure  as  the 
same  chemical  substance.  Thus,  ice,  water,  and  water-vapor, 
or  rhombic  sulphur,  monosymmetric  sulphur,  liquid  sulphur, 
and  sulphur-vapor,  are  designated  different  physical  states  of 
the  same  substance.  Ozone  and  oxygen,  or  paracyanogen 
and  cyanogen,  are,  on  the  other  hand,  appropriately  regarded 
as  distinct  substances ;  for,  though  increase  of  temperature 
will  convert  the  former  into  the  latter,  the  oxygen  so  pro- 
duced can  not  be  changed  back  to  ozone,  or  the  cyanogen  to 
paracyanogen,  merely  by  restoring  the  original  temperature. 

10.  Elementary  and  Compound  Substances.  —  When 
chemical  substances  are  brought  together,  or  are  subjected, 
alone  or  mixed,  to  a  high  temperature  or  to  certain  other 
influences,  a  change  often  takes  place  which  results  in  the 
formation  of  new  chemical  substances.  When  the  weight 
of  one  of  the  new  substances  formed  is  less  than  that  of 
the  substance  from  which  it  is  produced,  and  the  change  is 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.  33 

known  to  be  complete,  the  original  substance  is  said  to  have 
been  decomposed.  Now,  almost  all  chemical  substances 
can  be  made  to  undergo  some  change  of  this  kind,  but  a  rela- 
tively small  number  have  been  discovered  which  do  not  give 
rise  in  any  complete  change  that  they  undergo  to  a  new 
substance  whose  weight  is  less  than  that  of  the  original 
substance.  These  substances,  which  every  means  at  our 
disposal  has  failed  to  decompose,  are  called  elementary  sub- 
stances. All  other  chemical  substances  can  directly  or  indi- 
rectly be  produced  from,  or  resolved  into,  these  elementary 
ones  ;  they  are  therefore  considered  to  be  composed  of  them, 
and  are  called  compound  substances,  or  simply  compounds. 

ii.  Law  of  the  Conservation  of  the  Elements. — 
Certain  other  important  limitations  in  regard  to  the  trans- 
formation of  chemical  substances  into  one  another  are  also 
to  be  considered  in  this  connection.  In  the  first  place,  it  is 
found  that,  although  in  certain  cases  one  elementary  sub- 
stance is  transformed  into  another,  for  example,  oxygen  into 
ozone  or  diamond  into  graphite,  yet  there  exist  about  eighty 
elementary  substances  or  groups  of  elementary  substances 
which  are  not  transformed  into  one  another  by  any  process 
whatever,  whether  direct  or  indirect.  Thus  the  innumer- 
able efforts  of  the  alchemists  and  the  experiments  of  modern 
investigators  with  their  vastly  greater  resources  have  alike 
failed  to  transform  copper  into  gold,  lead  into  silver,  or 
chlorine  into  oxygen,  even  with  the  help  of  indirect  pro- 
cesses or  of  those  of  a  most  energetic  character.  Corres- 
pondingly, compounds  or  mixtures  of  them  are  never 
transformed  except  into  such  other  compounds  or  mixtures 
as  can  be  decomposed  into  the  same  elementary  substances 
as  can  the  original  compounds ;  thus  a  mixture  of  lead 
chloride  and  potassium  bromide  is  readily  transformed  into 
one  of  lead  bromide  and  potassium  chloride,  but  it  is  impos- 
sible to  prepare  from  such  a  mixture  a  silver  salt  or  an 
iodide.  Furthermore,  when  two  or  more  elementary  sub- 
stances unite  to  form  compounds,  or  mixtures  of  them,  even 


34  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

though  almost  all  of  the  properties  of  the  elementary  sub- 
stances may  have  disappeared,  it  is  always  possible  by  suit- 
able methods  to  get  back  exactly  the  same  amounts  of  each 
of  the  elementary  substances  as  originally  entered  into  the 
compounds. 

These  facts  have  led  to  the  assumptions  that  there 
exist  constant  unchangeable  amounts  of  a  comparatively 
small  number  of  distinct  kinds  or  forms  of  matter,  and  that 
elementary  substances  contain  only  one  of  these,  while  com- 
pound substances  contain  two  or  more  of  them.  These 
distinct  kinds  or  forms  of  matter  are  called  elements ;  and 
the  empirically  established  principle  that  they  are  neither 
created  nor  destroyed  by  any  process  whatever  is  appropri- 
ately called  the  Law  of  the  Conservation  of  the  Elements. 

The  abstract  conception  denoted  by  the  word  element 
will  be  seen  to  have  a  significance  distinct  from  that  of  the 
concrete  term  elementary  substance.  Thus  it  would  be  inap- 
propriate to  speak  of  the  existence  of  elementary  substances 
in  compounds  ;  for  hardly  any  of  their  properties  are,  as  a  rule, 
exhibited  by  the  compounds.  Nor  are  the  properties  of 
either  elementary  or  compound  substances  determined  solely 
by  the  elements  contained  in  them ;  their  properties  are 
rather  to  be  regarded  as  the  result  of  the  temporary  associa- 
tion with  the  element  or  elements  of  quantities  of  energy. 
Moreover,  it  is  not  to  be  inferred  from  the  use  of  the  ex- 
pression "distinct  kinds  or  forms  of  matter"  in  the  defini- 
tion of  the  elements  that  there  exist  eighty  or  more 
entirely  unrelated  ultimate  realities,  for  it  might  be  that  the 
elements  represent  only  certain  permanent  associations  of 
matter  and  energy :  the  ultimate  nature  of  the  elements, 
like  that  of  matter,  is  entirely  unknown.  The  concept  of 
elements  and  the  assumption  of  their  existence  in  both 
elementary  and  compound  substances  furnish  convenient 
means,  avoiding  much  circumlocution,  of  expressing  the  fact, 
stated  in  the  first  paragraph  of  this  section,  that  the  quantity 
of  any  elementary  substance  which  can  be  obtained  from 


GENERAL   PRINCIPLES   RELATING    TO   MATTER.          36 

any  body  is  constant,  whatever  changes  that  body,  or  the 
chemical  substances  composing  it,  may  undergo.  The  term 
element  further  implies,  to  be  sure,  the  permanent  existence 
of  a  real  thing,  whose  nature  is  unknown,  that  gives  rise  to 
this  constancy.  Thus  its  significance  is  entirely  similar  in 
kind  to  that  of  the  term  matter. 

A  list  of  the  well-established  elements,  with  the  sym- 
bols by  which  they  are  represented,  is  given  in  §  16.  Others 
undoubtedly  exist,  for  new  ones  are  frequently  discovered ; 
and  some  few  of  those  now  regarded  as  such  may  prove  to 
be  mixtures  or  compounds.  Indeed,  it  is  impossible  to  say 
that  some  process  may  not  be  discovered  by  which  all  the 
present  elementary  substances  can  be  resolved  into  simpler 
substances.  The  existence,  however,  of  certain  similarities 
and  relationships  between  the  properties  of  the  various  ele- 
mentary substances  and  of  certain  systematic  differences  be- 
tween their  properties  and  those  of  compound  substances, 
and  the  relative  permanence  of  the  elements  (even  if  it 
should  prove  not  to  be  an  absolute  one,  as  all  our  experience 
thus  far  indicates  it  to  be),  furnish  conclusive  proof  that  all 
the  well-known  elements  are  substances  of  the  same  order, 
and  of  an  order  distinct  from  that  to  which  our  present 
compound  substances  belong.  This  is  not  the  place  to 
present  in  detail  the  facts  which  have  led  to  this  conclusion, 
but  it  may  be  mentioned  that  among  the  most  important  of 
them  are  the  relationships  between  the  properties  of  elemen- 
tary substances  which  are  brought  out  by  the  so-called 
Periodic  System,  and  the  uniformity  in  the  heat-capacities 
of  solid  elementary  substances  which  is  expressed  by  the 
Law  of  Dulong  and  Petit. 

12.  Law  of  Definite  Proportions.  —  Investigations 
of  the  elementary  composition  of  substances  have  proved 
that  a  series  of  pure  chemical  compounds  containing  the 
component  elements  in  continuously  varying  proportions  by 
weight  does  not  exist,  but  that  there  exists  only  a  finite 
number  of  compounds,  each  of  perfectly  definite  composi- 


36  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

tion.  Chemical  substances  obtained  from  different  sources 
or  by  different  methods  are  either  absolutely  identical  in 
composition,  or  they  differ  from  one  another  by  consider- 
able amounts.  When  a  chemical  substance  changes  in 
composition,  even  though  the  change  may  take  place  grad- 
ually, it  is  never  possible  to  isolate  by  the  methods  illus- 
trated in  §  9  a  series  of  chemical  substances  varying 
continuously  in  composition,  but  the  change  results  in  the 
formation  of  a  single  new  substance  (or  a  small  number  of 
new  substances)  differing  markedly  in  composition  from  the 
original  substance  (and  from  one  another).  There  is  never 
a  gradual  transition  in  the  composition  of  chemical  com- 
pounds, but  always  a  sharp,  sudden  change.  These  state- 
ments are  an  expression  of  what  is  known  as  the  Law  of 
Definite  Proportions. 

For  example,  when  salt  is  prepared  from  sodium  hy- 
droxide and  hydrochloric  acid,  it  is  not  found  that  its  com- 
position is  in  any  way  dependent  on  the  proportions  of  the 
ingredients  taken  or  on  the  conditions  of  temperature  or 
concentration ;  the  product  is  not  found  to  contain  a  larger 
proportion  of  chlorine  when  an  excess  of  hydrochloric  acid 
is  used,  nor  of  sodium  when  an  excess  of  sodium  hydroxide 
is  employed.  When  zinc  is  burnt,  there  are  no  intermediate 
products  between  it  and  its  oxide ;  but  if  the  combustion 
is  interrupted  before  its  completion,  the  product  is  found 
to  be  a  mixture  of  unchanged  zinc  and  its  oxide,  not  a 
single  chemical  compound  intermediate  in  composition  and 
properties. 

The  Law  of  Definite  Proportions,  which  had  before 
been  tacitly  recognized,  was  energetically  disputed  at  the 
beginning  of  the  last  century,  but  was  soon  established  by 
careful  analyses  with  as  great  a  degree  of  accuracy  as  the 
state  of  analytical  chemistry  then  permitted.  More  recently 
it  was  suggested  that  the  composition  of  chemical  compounds 
might  still  be  variable  within  narrow  limits,  but  the  experi- 
ments of  Stas,  made  with  a  view  of  testing  this  supposition, 


GENERAL  PRINCIPLES  RELATING    TO  MATTER.          37 

showed  that  the  variations,  if  any  exist,  are  less  than  the 
unavoidable  experimental  errors  of  analysis.  For  he  de- 
termined, in  the  case  of  five  distinct  samples  of  ammonium 
chloride  obtained  from  entirely  different  sources  or  by 
different  methods  of  purification,  the  weight  of  silver  which, 
after  solution  in  nitric  acid,  sufficed  to  precipitate  a  definite 
weight  of  the  chloride,  and  found  the  extreme  variations 
in  the  ratio  of  these  weights  to  be  less  than  one  ten-thou- 
sandth part  of  its  value. 

It  may  be  added  that  all  the  properties  of  pure  chemi- 
cal substances  exhibit  the  same  definiteness  and  discon- 
tinuity as  does  their  elementary  composition,  and  that  the 
differentiation  of  chemical  substances  and  mixtures  finds 
its  experimental  justification  in  the  fact  that  there  exist  a 
limited  number  of  substances  with  perfectly  definite  prop- 
erties of  such  a  relatively  high  degree  of  stability  towards 
resolving  agencies  that  they  can  be  prepared  with  the  help 
of  such  agencies  from  a  variety  of  materials.  The  Law  of 
Definite  Proportions  and  the  principle  of  the  definiteness 
of  properties  in  general  are  therefore  involved  in  the 
definitions  of  chemical  substances  given  in  §  9. 

13.  The  Law  of  Multiple  Proportions.  —  Investiga- 
tions have  also  shown  that  certain  simple  relations  exist 
between  the  quantities  of  the  elements  combined  with  one 
another  in  different  chemical  compounds.  These  are  com- 
monly expressed  in  the  form  of  two  principles,  called  the 
Law  of  Multiple  Proportions  and  the  Law  of  Combining 
Weights. 

The  Law  of  Multiple  Proportions  is  as  follows  :  When 
one  element  combines  with  another  in  several  proportions  to 
form  different  chemical  compounds,  the  quantities  of  the  one 
element  which  in  the  several  compounds  are  combined  with 
the  same  quantity  of  the  other  element,  stand  to  one  another 
in  the  ratio  of  small  whole  numbers. 

For  example,  in  one  of  the  two  known  compounds  of 
carbon  and  oxygen,  16  parts  by  weight  of  oxygen,  and  in 


38  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

the  other  32  parts  of  it,  are  combined  with  12  parts  of  car- 
bon. Five  compounds  of  nitrogen  and  oxygen  exist,  in 
which  the  quantities  of  the  latter  combined  with  any  defi- 
nite quantity  of  the  former  are  to  one  another  exactly  as 
1:2:3:4:5.  Sometimes,  however,  especially  in  the  case 
of  compounds  of  the  element  carbon,  the  ratio  is  much  less 
simple ;  thus,  in  the  two  hydrocarbons,  naphthalene  and 
anthracene,  the  relative  quantities  of  hydrogen  combined 
with  one  gram  of  carbon  are  as  28  :  25. 

14.  The  Law  of  Combining  Weights.  —  Careful 
investigations  of  the  quantitative  elementary  composition  of 
chemical  substances  have  shown  that  to  the  elements  in- 
dividually can  be  assigned  definite  numerical  values  which 
accurately  express  the  weights  of  them,  or  small  multiples 
of  the  weights  of  them,  which  are  combined  with  one 
another  in  all  chemical  compounds.  Such  numerical  values 
are  called  the  combining  weights  of  the  elements.  Since 
they  are  essentially  relative  quantities,  greater  definiteness 
can  be  attained  by  adopting  a  definite  weight  of  some  one 
element  as  a  standard  of  reference.  Adopting  16  grams  of 
oxygen  as  the  standard  in  accordance  with  what  is  probably 
the  best  practice,  the  combining  weight  of  an  element  is 
defined  to  be  that  weight  of  it  which  combines  with  16 
grams  of  oxygen,  or  with  some  small  multiple  or  submultiple 
of  1 6  grams  of  oxygen.  And  with  the  help  of  this  definition 
the  principle  involved  in  the  preceding  statement,  and  known 
as  the  Law  of  Combining  Weights,  can  be  concisely  expressed 
as  follows  :  Elements  combine  with  one  another  only  in  the 
proportions  of  their  combining  weights  or  of  small  multiples 
of  them. 

This  law  may  be  illustrated  by  citing  the  experiments 
of  Stas  which  have  furnished  the  most  exact  confirmation 
of  it.  By  dissolving  carefully  purified  silver  chlorate  in 
water,  reducing  it  to  the  insoluble  chloride,  and  testing  the 
solution  for  an  excess  of  silver  and  chlorine,  he  proved  that 
the  ratio  of  the  quantities  of  these  two  elements  in  silver 


GENERAL  PRINCIPLES  RELATING    TO  MATTER.          39 

chlorate  does  not  differ  from  the  ratio  of  them  in  silver 
chloride  by  more  than  one  ten-millionth  part  of  its  value, 
which  was  the  limit  of  delicacy  of  the  tests.  Similar  ex- 
periments were  made  with  similar  results  with  silver  bro- 
mate  and  iodate.  Careful  determinations  were  also  made,  by 
indirect  methods,  of  the  quantities  of  silver  combined  with  a 
definite  quantity  of  oxygen  in  these  three  salts  and  in  silver 
sulphate.  The  quantities  combined  with  16  grams  of  oxygen 
were  found  to  be  as  follows  : 

In  silver  chlorate 35.980  grams. 

In  silver  bromate 35-974  grams. 

In  silver  iodate 35.979  grams. 

In  silver  sulphate    ....     53.963  =  f  X  35-976  grams. 

It  will  be  seen  that  the  numerical  values  adopted  as 
the  combining  weights  are,  so  far  as  the  Law  of  Combining 
Weights  is  concerned,  arbitrary  in  two  respects :  first,  be- 
cause it  is  necessary  to  adopt  an  arbitrary  standard  of  refer- 
ence ;  and  secondly,  because,  in  the  case  of  each  element, 
some  arbitrary  multiple  of  the  value  obtained  by  the  analysis 
of  any  one  of  its  compounds  must  be  selected.  A  decision 
in  regard  to  both  of  these  matters  must  be  reached  by 
general  agreement,  but  it  should  be  such  as  will  secure 
the  greatest  degree  of  permanence  in  the  system  of  com- 
bining weights  adopted,  and  will  furnish  the  simplest  ex- 
pression of  the  elementary  composition  and  allied  proper- 
ties of  chemical  substances. 

The  reasons  for  adopting  16  grams  of  oxygen  as  the 
standard  of  reference  may  first  be  briefly  considered.  In- 
stead of  oxygen,  hydrogen  was  for  a  long  time  almost  uni- 
versally employed  as  the  standard  substance,  and  it  still 
continues  to  be  used  to  a  considerable  extent.  It  has  the 
advantage  of  possessing  the  smallest  combining  weight,  so 
that  a  unit-weight  (one  gram)  of  it  can  be  adopted  as  the 
standard  weight  without  making  it  necessary  to  express  the 
combining  weights  of  other  elements  in  the  form  of  incon- 


40  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

-venient  fractions.  Hydrogen  has  also  another,  more  impor- 
tant advantage  over  oxygen,  by  reason  of  the  fact  that,  in 
comparing  the  quantities  of  different  chemical  substances 
which  enter  into  reactions  with  one  another,  it  is,  as  a  rule, 
simpler  and  more  in  accordance  with  actually  existing  rela- 
tions to  use  as  the  basis  of  comparison  the  quantity  of  hydro- 
gen rather  than  the  quantity  of  oxygen  involved  in  the 
reactions  in  question  or  in  similar  ones,  —  a  fact  which  will 
come  into  more  prominence  in  connection  with  the  defini- 
tion of  equivalent  weights  (§19).  Oxygen,  on  the  other 
hand,  has  the  great  practical  advantage  that  the  combining 
weights  of  almost  all  the  other  elements  have  been  actually 
determined  either  from  the  composition  of  their  oxides  or 
from  that  of  their  compounds  with  the  halogens,  the  com- 
bining weights  of  which  latter  have  themselves  been  estab- 
lished by  determining  the  relative  quantities  of  halogen  and 
oxygen  present  in  compounds  containing  both  of  them  (see 
§  1 5).  The  experimental  basis  of  the  system  of  combining 
weights  is  therefore  oxygen,  and  not  hydrogen ;  and  if  the 
latter  were  adopted  as  the  conventional  basis,  it  would  be 
necessary  to  calculate  the  values  over  to  the  hydrogen  stand- 
ard with  the  help  of  the  ratio  between  the  combining  weights 
of  hydrogen  and  oxygen.  Every  new,  more  accurate  deter- 
mination of  this  ratio  would  then  involve  a  corresponding 
change  in  the  values  of  almost  all  the  other  combining 
weights  :  moreover,  until  very  recently,  a  great  degree  of  un- 
certainty has  existed  in  regard  to  the  exact  value  of  this  ratio  ; 
and,  even  at  the  present  time,  it  is  not  known  as  accurately 
as  the  ratio  of  the  combining  weights  of  some  of  the  other 
elements  to  that  of  oxygen.  For  these  reasons,  although 
some  difference  of  opinion  does  still  exist  in  regard  to  the 
relative  importance  of  the  different  advantages  which  the 
two  elements  possess,  it  seems  at  present  to  be  the  best 
practice  to  refer  the  combining  weights  to  oxygen,  and  this 
practice  will  be  adhered  to  throughout  this  work.  The 
principal  reasons  for  adopting  16  grams  of  this  element 


GENERAL   PRINCIPLES  RELATING    TO  MATTER.          41 

(instead  of  one  gram,  for  example),  as  the  standard  of 
reference,  are  :  first,  that  it  involved  a  comparatively  slight 
change  in  the  values  of  the  combining  weights  from  those 
prevailing  some  years  ago,  when  the  commonly  employed 
hydrogen  standard  was  to  be  replaced ;  and  second,  that  the 
combining  weight  of  hydrogen  is  still  not  far  from  unity 
(1.0075,  according  to  the  most  accurate  determinations),  so 
that  to  some  extent  at  least  —  thus,  in  cases  where  the  great- 
est accuracy  is  not  required — the  advantages  of  the  hydrogen 
standard  are  retained. 

In  regard  to  the  adoption  of  definite  multiples  of  the 
values  obtained  by  chemical  analysis,  it  can  only  be  stated 
at  this  point  that  investigators,  guided  by  certain  theoretical 
considerations,  have  reached  a  substantially  unanimous  agree- 
ment, and  have  succeeded  in  establishing  a  system  of  com- 
bining weights  which  not  only  serves  to  express  in  a  simple 
manner  the  elementary  composition  of  chemical  substances, 
but  also  forms  a  highly  satisfactory  basis  for  the  develop- 
ment of  the  knowledge  of  their  properties  and  transforma- 
tions. It  should  be  distinctly  understood,  however,  that  the 
conception  of  combining  weights  does  not  involve  the  use 
of  these  particular  multiples  or  of  any  definite  multiples  of 
the  analytical  values ;  and  if,  for  the  sake  of  convenience, 
such  are  adopted,  it  must  be  fully  realized  that  the  multiply- 
ing factors  are  arbitrary  ones  in  the  sense  that  they  can  not 
be  determined  by  quantitative  analyses  of  chemical  sub- 
stances. 

The  method  of  calculating  combining  weights  from 
analytical  data  and  the  arbitrariness  of  adopting  a  definite 
multiple  may  be  illustrated  by  examples.  Syntheses  of 
carbon  dioxide  from  its  elements  have  shown  that  it  contains 
27.27  per  cent,  of  carbon  and  72.73  per  cent,  of  oxygen. 
The  combining  weight  of  carbon  referred  to  that  of  oxygen 

taken  as  16.00  is  therefore  2''2'  ^  16.00  =  6.00,  or  any 
small  multiple  or  submultiple  of  this  value.  Carbon  monox- 


42  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

ide  is  found  to  contain  42.50  per  cent,  of  carbon  and  57.50 
per  cent,  of  oxygen,  from  which  it  follows  that  the  combin- 
ing weight  is  —  -  x  16.00  =  12.00,  or  any  small  multiple 

or  submultiple  of  it.  Finally,  alcohol  is  found  by  analysis 
to  contain  52.12  per  cent,  of  carbon,  34.75  per  cent,  of 
oxygen,  and  13.14  per  cent,  of  hydrogen;  the  combining 

weight  of  carbon  is,  therefore,  — '. —  x    16.00  =  24.00,  or 

34-75 

any  small  multiple  or  submultiple  of  it.  It  would  evidently 
be  entirely  arbitrary  to  select  as  the  combining  weight  a 
particular  one  of  the  three  values  calculated  directly  from 
the  composition  of  these  different  compounds. 

It  should  be  added  that  the  term  combining  weight  is 
often  used  in  a  similar  sense  in  connection  with  compound 
substances.  The  combining  weight  of  a  compound  is  defined 
to  be  that  weight  of  it  which  contains  such  weights  of  the 
component  elements  as  are  their  combining  weights  or  small 
multiples  of  them. 

In  closing  this  consideration  of  the  Law  of  Combining 
Weights  attention  may  be  called  to  the  fact  that  the  Law 
includes  as  special  cases  the  Laws  of  Definite  and  Multiple 
Proportions.  It  is  therefore  the  general  law  relating  to  the 
elementary  composition  of  pure  chemical  substances. 

15.  Determination  of  the  Combining  Weights.  — 
The  combining  weights  of  the  elements  are  constants  of 
so  great  importance  that  a  general  knowledge  of  the  way 
in  which  they  have  been  determined  is  desirable.  A  partial 
outline  of  the  extensive  and  highly  accurate  series  of  deter- 
minations which  were  made  by  Stas  in  the  years  1860  to 
1864  will  therefore  be  presented.  This  will  not  only  serve 
to  illustrate  the  general  character  of  the  methods  of  deter- 
mining combining  weights,  but  will  also  give  a  good  idea 
of  the  experimental  basis  upon  which  the  commonly  em- 
ployed system  of  combining  weights  in  large  part  rests. 

In  determining  the  combining  weight  of  silver  the  fol- 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.          43 

lowing  five  methods  were  employed  :  i.  A  known  weight  of 
pure  potassium  chlorate  was  converted,  in  some  experiments 
by  ignition,  and  in  others  by  treatment  with  hydrochloric 
acid,  into  potassium  chloride,  and  the  weight  of  the  latter 
compound  formed  accurately  determined.  The  combining 
weight  of  the  potassium  chloride  referred  to  oxygen  is  then 
evidently  given  by  the  proportion,  the  weight  of  the  potas- 
sium chloride  obtained  is  to  the  difference  between  the 
weight  of  the  potassium  chlorate  taken  and  that  of  the 
potassium  chloride  obtained  as  the  combining  weight  of 
potassium  chloride  is  to  the  combining  weight  of  oxygen 
(assumed  to  be  1 6  or  some  multiple  of  it).  The  quantity 
of  oxygen  in  potassium  chlorate  is  assumed  to  be  48,  that 
is,  three  times  the  combining  weight,  since  it  has  been 
found  that  the  composition  of  chemical  compounds  is,  on 
the  whole,  more  simply  expressed  when  this  assumption  is 
made.  Stas  used  801.4780  grams  of  potassium  chlorate  in 
eight  experiments,  which  gave,  upon  reduction,  487.6605 
grams  of  potassium  chloride,  whence  follows  a  value  of 
74.5902  ±  0.0045  f°r  tne  combining  weight  of  the  latter. 
(The  quantity  following  the  ±  sign  signifies  here,  as  usual, 
the  probable  error  of  the  result.)  Weighed  quantities  of 
potassium  chloride  and  metallic  silver  were  then  dissolved 
in  water  and  dilute  nitric  acid,  respectively,  and  the  rela- 
tive quantities  of  the  two  -substances  which  exactly  precip- 
itated each  other  determined.  The  sum  of  the  amounts 
of  potassium  chloride  used  in  twenty-three  experiments  was 
found  to  be  147.70775  grams,  and  to  this  210.85508  grams 
of  silver  were  found  to  correspond ;  and  from  the  propor- 
tion, 147.70775  :  210.85508  :  :  74.5902  :  the  combining 
weight  of  silver,  follows  for  the  last  named  quantity  the 
value  107.9401  d-  0.0058. 

2.  Weighed  quantities  of  silver  chlorate  were  dissolved 
in  water  and  reduced  by  means  of  sulphurous  acid  to  silver 
chloride.  The  composition  of  the  latter  was  also  determined, 
first,  by  heating  silver  in  chlorine  gas,  and  second,  by  dis- 


44  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

solving  it  in  nitric  acid  and  precipitating  with  hydrochloric 
acid  or  ammonium  chloride. 

3.  Silver  bromate  was  reduced  in  the  same  way  to  silver 
bromide,  in  order  to  determine  the  combining  weight  of  the 
latter  with  reference  to  that  of  oxygen.     The  composition 
of  the  silver  bromide  was  determined  by  a  complete  synthesis  ; 
that  is,  the  weights  of  silver  and  of  bromine  which  combine 
with  each  other  were  determined,  as  well  as  the  weight  of 
their  product.      The  fact  that  the  last  weight  was  almost 
exactly  equal  to  the  sum  of  the  weights  of   the  elements 
taken  is  an  almost  conclusive  proof  of  the  purity  of  all  the 
substances,  and  of  the  accuracy  of  the  experiments. 

4.  The  combining  weight  of  silver  iodide  was  similarly 
determined  by  reducing  silver  iodate  with  sulphurous  acid 
solution  to  silver  iodide,  and  also  by  a  complete  analysis  of 
the  iodate,  which  consisted  in  decomposing  weighed  quanti- 
ties of  it  by  heat  and  weighing  the  quantities  of  silver  iodide 
and  oxygen  produced.     The  composition  of  silver  iodide  was 
determined  by  a  complete  synthesis. 

5.  Silver  sulphate  was    reduced  to  metallic  silver  by 
continued  heating  in  a  stream  of  hydrogen  gas.     The  com- 
position of  silver  sulphide  was  also  determined. 

The  results  pbtained  by  the  five  methods  which  have 
been  just  described  are  shown  in  the  following  table.  The 
numbers  are  the  combining  weights  of  the  substances  whose 
formulas  are  given  at  the  beginning  of  the  separate  lines. 

1.  KC1  from  KC1  :  KC1O3  =    74-59°2  ±  0.0045 

2.  AgCl  from  AgCl  :  AgClO3  =  143.3940  dz  0.0064 

3.  AgBr  from  AgBr  :  AgBrO3  =  187.8759  ±  0.0237 

4.  Agl  from  Agl  :  AgIO3  =  234.7907  ±  0.0095 

5.  Ag2S  from  |g|  |  ^  }      =  247-9O74  =b 


GENERAL   PRINCIPLES  RELATING    TO  MATTER.          45 

1.  Ag  from  Ag  :  KC1  =  107.9401  rb  0.0058 

2.  Ag  from  Ag  :  AgCl  =  107.9406  ±  0.0049 

3.  Ag  from  Ag  :  AgBr  =  107.9233  ±  0.0140 

4.  Ag  from  Ag  :  Agl  =  107.9371  ±  0.0045 

5.  Ag  from  2Ag  :  Ag2S  =  107.9270  db  0.0090 

Mean  =  107.9376  ±  0.0037 

From  the  results  just  tabulated  the  combining  weights 
of  chlorine,  bromine,  iodine,  and  sulphur  can  be  directly  ob- 
tained by  subtracting  the  mean  value  of  the  combining 
weight  of  silver  from  the  combining  weight  of  the  corre- 
sponding silver  compound,  and  that  of  potassium,  by  sub- 
tracting that  of  chlorine  from  the  combining  weight  of 
potassium  chloride.  The  combining  weights  of  the  first 
four  elements  can  also  be  computed  from  the  mean  value 
for  silver  and  the  composition  of  their  compounds  with  it. 
The  values  obtained  by  a  combination  of  these  two  methods 
are: 

Cl  =    35.4529  ±  0.0037 

Br  =    79.9628  rb  0.0032 

I     =  126.8640  db  0.0035 

S    =    32.0626  =b  0.0042 

K  =    39.1361  zb  0.0032 

In  a  similar  manner  the  combining  weights  of  other 
elements  have  been  determined.  One  of  the  most  common 
processes  consists  in  preparing  the  chloride  or  bromide  of 
the  element,  and  in  determining  the  proportion  of  chlorine 
or  bromine  in  it  by  precipitation  with  silver  nitrate. 

16.  Numerical  Values  of  the  Combining  Weights. — 
The  following  table  contains  the  names  of  the  well-estab- 
lished elements,  the  symbols  by  which  they  are  commonly 
represented,  and  the  most  probable  values  of  their  combin- 
ing weights,  which  have  been  derived  through  a  critical  con- 
sideration of  the  results  of  all  the  determinations  which  have 
been  made  up  to  the  present  time.  Those  multiples  of  the 
analytical  values  are  given  which  are  now  commonly  em- 


46  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

ployed  by  chemists,  and  which,  by  reason  of  certain  theo- 
retical considerations  connected  with  them,  are  designated 
atomic  weights.  The  values  are  referred  to  the  oxygen 
standard,  the  combining  weight  of  this  element  being  as- 
sumed to  be  16.000.  Such  a  number  of  figures  is  retained 
that  the  figure  before  the  last  is  probably  not  uncertain  by 
as  much  as  one  unit.  Columbium  and  niobium,  beryllium 
and  glucinum,  are  merely  different  names  for  the  same 
elements.  As  no  compounds  of  the  recently  discovered 
gaseous  elements,  argon,  helium,  krypton,  neon,  and  xenon, 
have  been  prepared  and  analyzed,  no  combining  weights 
are  assigned  to  them  in  the  table. 


GENERAL  PRINCIPLES  RELATING   TO  MATTER. 


47 


TABLE    OF   COMBINING   AND    ATOMIC    WEIGHTS. 


Aluminum     . 

Al 

27.1 

Neodymium  .     . 

Nd 

143.6 

Antimony  .     . 

Sb 

120.0 

Neon   .    .    .    V 

Ne 

... 

Argon  .     .     . 

A 

.  .  . 

Nickel.     .     .     . 

Ni 

58.70 

Arsenic      .     . 

As 

75.0 

Niobium  . 

Nb 

94. 

Barium      .     . 

Ba 

137.43 

Nitrogen  .     .     . 

N 

14.04 

Beryllium  .     . 

Be 

9.1 

Osmium   .     .     . 

Os 

190.8 

Bismuth    .     . 

Bi 

208. 

Oxygen    .     .    . 

0 

16.000 

Boron   .     .     . 

B 

11.0 

Palladium     i    . 

Pd 

106.5 

Bromine    .     . 

Br 

79.955 

Phosphorus  .    . 

P 

31.0 

Cadmium  .    . 

Cd 

112.3 

Platinum  .     .    ^. 

Pt 

195.2 

Caesium     .     . 

Cs 

132.9 

Potassium     .     . 

K 

39.14 

Calcium     .     . 

Ca 

40.1 

Praseodymium  . 

Pr 

140.5 

Carbon  .     .     . 

C 

12.001 

Rhodium  .    .     . 

Rh 

103.0 

Cerium  .     .     . 

Ce 

140. 

Rubidium    »..     ,. 

Rb 

85.44 

Chlorine    .     . 

Cl 

35.455 

Ruthenium  .     . 

Ru 

101.7 

Chromium 

Cr 

52.14 

Samarium     .    . 

Sm 

150. 

Cobalt  .     .     . 

Co 

59.00 

Scandium     .    . 

Sc 

44. 

Columbium    . 

Cb 

94. 

Selenium  .     .     . 

80 

79.2 

Copper  .     .    . 

Cu 

63.604 

Silicon.    .    .    . 

Si 

28.4 

Erbium     .     . 

Er 

166. 

Silver  .... 

Ag 

107.93 

Fluorine    .     . 

El 

19.05 

Sodium     .    .    . 

Na 

23.05 

Gadolinium   . 

Gd 

156. 

Strontium     .     . 

Sr 

87.68 

Gallium     .     . 

Ga 

70.0 

Sulphur    .     .     . 

S 

32.065 

Germanium   . 

Ge 

72.5 

Tantalum     .    * 

Ta 

183. 

Glucinum  .    . 

Gl 

9.1 

Tellurium     .    . 

Te 

127.5 

Gold      .     .     . 

Au 

197.3 

Terbium  .    .     . 

Tb 

160. 

Helium      .     . 

He 

.  .  • 

Thallium  .    .    . 

Tl 

204.15 

Hydrogen  .     . 

H 

1.0075 

Thorium  .     .     . 

Th 

233. 

Indium      .     . 

In 

114. 

Thulium  .     .     . 

Tu 

170. 

Iodine  .     . 

I 

126.85 

Tin.    .    .     .     . 

Sn 

119.0 

Indium     .    . 

Ir 

193.0 

Titanium  .    .     . 

Ti 

48.17 

Iron.     .     .     . 

Ee 

55.9 

Tungsten      .    . 

W 

184. 

Krypton    .    . 

Kr 

Uranium  .     .     . 

U 

238.5 

Lanthanum   . 

La 

138.5 

Vanadium    .     . 

V 

51.4 

Lead     .    .    . 

Pb 

206.92 

Xenon      .     .     . 

X 

... 

Lithium     .     . 

Li 

7.03 

Ytterbium    .    . 

Yb 

173. 

Magnesium    . 

Mg 

24.36 

Yttrium   .    .    . 

Y 

89.0 

Manganese    . 

Mn 

55.02 

Zinc     .... 

Zn 

65.40 

Mercury    .     . 

Hg 

200.0 

Zirconium     .    . 

Zr 

90.6 

Molybdenum 

Mo 

96.0 

48  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

17.  Elementary  Composition  as  a  Means  of  Distin- 
guishing Chemical  Substances  from  Mixtures.  —  Since 
pure  chemical  substances  contain  the  elements  in  quantities 
that  are  proportional  to  their  combining  weights  or  small 
multiples  of  them,  while  mixtures  may  contain  the  elements 
in  any  proportion,  it  is  evident  that  a  knowledge  of  the 
quantitative  elementary  composition  of  a  substance  is  of  as- 
sistance in  determining  whether  it  is  a  pure  compound  or  a 
mixture.  It  is  not  true,  however,  as  may  seem  at  first  to  be 
the  case,  that  a  correct  decision  can  always  be  reached  by 
this  method.  For  this  there  are  two  reasons.  The  first 
is  that  the  analyses  of  substances  proved  to  be  pure  by 
the  two  methods  described  in  §  9  have  shown,  that,  while 
in  the  case  of  the  compounds  of  almost  all  the  elements 
the  small  whole  numbers  by  which  the  combining  weights 
must  be  multiplied  in  order  to  express  the  composition  cor- 
rectly, seldom  exceed  a  few  units  in  value,  yet  in  the  case  of 
some  of  the  compounds  of  a  few  elements,  notably  carbon, 
the  values  of  these  multiplying  factors  are  so  large,  that,  in 
view  of  the  unavoidable  analytical  errors,  the  elementary 
composition  is  of  no  assistance  in  distinguishing  such  com- 
pounds from  mixtures.  The  second  reason  is  that  experi- 
ence has  shown  that  isomeric  substances  exist,  that  is,  sub- 
stances which,  though  distinct  in  some  of  their  other  proper- 
ties, are  identical  in  their  elementary  composition.  It  may 
be  added  that  isomeric  substances,  like  those  of  complex 
composition,  are  extremely  common  among  the  compounds 
of  carbon,  but  are  comparatively  rare  among  those  not  con- 
taining this  element. 

Conformity  in  composition  with  that  required  by  the 
Law  of  Combining  Weights  is  therefore  only  an  important 
indication,  not  a  universally  applicable  criterion,  of  a  pure 
chemical  substance,  while  apparent  non-conformity  with  it 
is  a  strong  indication,  but  not  conclusive  evidence  of  a  mix- 
ture. The  indications  are  generally  reliable  in  the  case  of 
the  compounds  of  other  elements  than  carbon,  but  very 
uncertain  in  the  case  of  these. 


GENERAL   PRINCIPLES  RELATING     TO  MATTER.  49 

Some  examples  of  non-conformity  with  the  Law  may  be 
presented.  If  a  sample  of  rock-salt  were  found  on  analysis 
to  contain,  in  combination  with  35.45  grams  of  chlorine, 
22.50  grams  of  sodium,  instead  of  the  23.05  grams  which 
its  combining  weight  requires,  the  salt  would  be  correctly 
pronounced  a  mixture.  But  if  the  pure  substance  stearine, 
which  in  reality  contains  the  elements  carbon,  hydrogen,  and 
oxygen,  in  the  ratio  of  57  combining  weights  of  the  first  to 
no  of  the  second  to  6  of  the  third,  were  accurately  analyzed, 
and  the  same  reasoning  applied  to  the  results,  the  erroneous 
conclusion  that  it  also  is  a  mixture  might  be  drawn,  on 
account  of  the  apparent  non-conformity  in  composition 
with  the  Law  of  Combining  Weights. 

It  is  evident  from  these  considerations  that,  if  the  Law 
of  Combining  Weights  is  to  apply  to  all  chemical  compounds, 
the  restriction  in  the  statement  of  it  to  the  effect  that  the 
multiples  of  the  combining  weights  of  the  component  ele- 
ments are  small  ones,  must  be  broadly  interpreted.  It  is 
nevertheless  true,  that  the  significance  of  these  laws,  both 
from  an  analytical  and  a  theoretical  standpoint,  depends  very 
largely  on  the  fact  that  the  integral  factors  involved  do  not 
exceed  a  few  units  in  value  in  the  case  of  a  large  proportion 
of  compounds. 

18.  Chemical  Formulas  and  Chemical  Equations. — 
In  order  to  express  the  gravimetric  composition  of  com- 
pounds, the  symbols  of  the  elements  are  considered  to  rep- 
resent their  combining  weights,  and  are  written  in  sequence 
with  such  integers  as  subscripts  as  will  make  the  resulting 
formulas  express  the  proportions  by  weight  of  the  elements  in 
the  compounds.  Such  formulas,  which  express  nothing  more 
than  the  results  of  analysis,  are  called  empirical  formulas. 
Thus,  water  is  found  to  contain  hydrogen  and  oxygen  in  the 
proportion  of  1.0075  :  8,  and  since  the  combining  weights 
of  these  elements  are  1.0075  and  J6,  the  empirical  formula 
of  water  is  H2O. 

In  order  to  determine  the  formula  of  a  compound  from 


* 
MIM   KAPH   MIM    LIBRARY 

C HEM.  B LOG.     U.  C 


60  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

its  percentage  composition,  the  percentage  weight  of  each 
element  is  divided  by  its  combining  weight,  and  the  result- 
ing values,  which  evidently  express  the  relative  number  of 
combining  weights  of  the  separate  elements  in  the  com- 
pound, are  then  multiplied  by  such  a  factor  as  will  make 
them  all  whole  numbers,  or  so  nearly  equal  to  whole  num- 
bers that  the  deviation  is  not  greater  than  can  be  accounted 
for  by  the  experimental  error.  This  factor  can  commonly 
be  obtained  at  once  by  inspection,  especially  if  the  relative 
values  are  first  simplified  by  dividing  them  all  by  the 
smallest  one  of  them. 

For  example,  ferric  oxide  is  found  by  analysis  to  contain 
30.0  per  cent,  of  oxygen  and  70.0  per  cent,  of  iron.  The 
ratio  of  the  number  of  combining  weights  of  these  elements 
in  the  compound  is,  therefore,  £|  :  f£,  or  1.250  :  1.875,  or 
I  :  1.5.  The  empirical  formula  is,  therefore,  Fe2O3. — An 
actual  analysis  of  an  organic  substance  by  the  ordinary 
method  showed  the  following  percentage  composition  : 
C  :  51.60;  H  :  7.05 ;  O  :  41.35.  Therefore  the  ratio  of  the 
number  of  combining  weights  is  |1-M.  :  1>y0'^6  :  f|fff ,  or 
4.30  :  7.00  :  2.58,  or  1.67  :  2.71  :  i.oo,  or  5.00  :  8.13  :  3.00. 
Since  the  deviation  from  an  integer  of  the  calculated  num- 
ber of  combining  weights  of  hydrogen  (8.13)  would  corre- 
spond to  a  variation  of  about  o.  i  of  a  unit  in  its  percentage 
weight  (7.05),  and  since  an  experimental  error  of  this  mag- 
nitude is  usual  in  such  analyses,  no  significance  is  to  be 
attached  to  the  deviation,  and  the  conclusion  is  to  be  drawn 
that  the  composition  of  the  compound  is  represented  within 
the  experimental  error  by  the  empirical  formula  C6H8O3. 

In  order,  on  the  other  hand,  to  calculate  the  percentage 
composition  of  a  compound  from  its  formula,  the  number 
of  combining  weights  of  each  element  (as  shown  by  the 
subscripts  in  the  formula)  is  multiplied  by  the  corresponding 
combining  weight  itself,  and  each  product  is  then  divided  by 
the  sum  of  the  products.  The  result  gives  evidently  the 
fractional  quantities  of  the  various  elements  in  the  com- 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.  51 

pound.  Thus  the  formula  of  cane  sugar  is  ClaHMOn.  The 
ratio  of  the  weights  of  the  elements  contained  in  it  is,  there- 
fore, 12  X  12. oo  :  22  X  1.0075  :  ii  X  16.00,  or  144  :  22.2  : 
176;  and  the  percentage  of  carbon  is  1441^0aa.a^.417fl>  or 
42.08. 

In  order  to  express  the  relative  quantities  of  substances 
that  enter  into  and  are  produced  by  chemical  transforma- 
tions, their  formulas  are  considered  to  represent  not  only 
the  proportions  by  weight  of  the  elements  of  which  they 
are  composed,  but  also  absolute  weights  of  the  substances 
equal  to  the  sum  of  the  weights  represented  by  the  symbols 
of  the  elements  in  the  formulas.  This  weight  may  be 
designated  the  formula-weight  of  the  substance.  Thus  the 
formula  of  sulphuric  acid,  H2SO4,  represents  (2  X  1.0075) 
+  32.065  +  (4  X  16.000)  =  98.080  grams  of  sulphuric  acid ; 
and  that  of  sodium  hydroxide,  NaOH,  23.050  +  16.000  + 
i. 008  =  40.058  grams  of  sodium  hydroxide.  The  formulas 
of  substances  involved  in  a  definite  reaction  (that  is,  in  a 
definite  chemical  transformation)  are  then  written  together 
in  the  form  of  an  equation,  each  formula  being  preceded 
by  such  an  integral  coefficient  as  will  make  the  equation 
express  the  relative  quantities  of  the  reacting  substances. 
For  example,  the  expression, 

H2S04  +  2NaOH  =  N^SC^  +  2H2O, 

signifies  that  98.080  parts  by  weight  of  sulphuric  acid  react 
with  80. 1 1 5  parts  of  sodium  hydroxide  with  the  production 
of  142.165  parts  of  sodium  sulphate  and  36.030  parts  of 
water. 

It  is  evidently  a  simple  matter  to  determine  with  the 
help  of  such  equations  the  quantity  of  a  product  obtainable 
from  a  definite  quantity  of  another  substance.  Two  con- 
siderations which  are  apt  to  be  overlooked  by  those  un- 
familiar with  such  calculations  may  be  referred  to,  however. 
In  the  first  place,  it  is  of  course  essential  that  the  formulas 
used  in  the  calculations  represent  the  composition  of  the 


62  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

substances  actually  weighed ;  for  example,  the  reaction  be- 
tween barium  chloride  and  potassium  sulphate  is  commonly 
written  as  follows  : 

BaCl2  +  K2SO4  =  BaSO4  +  2KC1 ; 

but,  if  pure  crystallized  barium  chloride,  which  contains 
water  of  crystallization,  is  the  substance  weighed  out,  its 
formula,  BaCl2.2H2O,  and  not  the  simpler  one  used  in  the 
above  equation,  must  be  employed  in  the  calculation. 
Secondly,  where  one  substance  is  converted  into  another  by 
a  series  of  chemical  reactions,  it  is  not  necessary  to  cal- 
culate the  quantities  of  the  intermediate  products  formed, 
but  it  suffices  to  compare  the  weights  corresponding  to  the 
formulas  of  the  original  and  final  substances ;  in  such  cases, 
however,  care  must  be  taken  to  use  in  the  calculation 
those  multiples  of  the  formula  weights  which  contain  equal 
weights  of  the  common  element.  For  example,  if  it  be 
desired  to  find  how  much  silver  chloride  is  formed  from  a 
given  weight  of  chlorine,  when  the  transformation  takes 
place  in  accordance  with  the  equations, 

H2  +  C12  =  2HC1,  and, 
AgN03  +  HC1  =  AgCl  +  HNO8, 

it  is  superfluous  to  calculate  first  the  quantity  of  hydro- 
chloric acid,  and  then  that  of  silver  chloride ;  for  the  lat- 
ter is  obtained  directly  from  the  ratio  2AgCl  :  C^  (not, 
however,  from  the  ratio  AgCl :  C12).  If  ferric  oxide  (Fe2O3) 
is  completely  transformed,  by  the  removal  of  oxygen  in  any 
manner  whatever,  into  ferroso-ferric  oxide  (Fe3O4),  the 
relative  weights  of  the  two  substances  are  not  expressed  by 
the  simple  ratio  Fe2O3  :  Fe3O4,  but  by  the  ratio  3Fe2O3  : 
2Fe3O4,  since  only  in  the  latter  case  do  the  two  formulas 
represent  quantities  of  the  substances  containing  the  same 
amount  of  iron. 

19.  Definition  of  Equivalent  Weights.  —  The  equa- 
tions used  as  examples  in  the  last  section  illustrate  the  fact 
that  the  relative  weights  of  the  substances  involved  in  any 


GENERAL  PRINCIPLES  RELATING    TO  MATTER.          63 

reaction  are  not  necessarily  the  formula-weights,  but  are 
frequently  small  multiples  of  these.  It  is  therefore  highly 
desirable  to  have  different  terms  for  these  different  quan- 
tities, and  it  has  become  customary  to  designate  the 
weights  of  substances  which  enter  into  reactions  with  one 
another,  or  take  the  place  of  one  another  in  corresponding 
reactions,  as  equivalent  weights.  In  giving  definiteness 
to  this  term,  the  difficulty  is  met  with  that  reactions  exhibit 
some  diversity  of  character,  and  that  those  weights  of  sub- 
stances which  are  comparable  with  one  another  with  respect 
to  one  kind  of  reactions  may  not  be  so  with  respect  to  other 
kinds.  This  difficulty  is  not  serious,  however,  in  the  case  of 
the  metallic  elements  and  their  compounds ;  for  most  of  the 
reactions  in  which  they  are  involved  are  included  in  one  of 
two  classes  known  respectively  as  metathetical  reactions, 
and  as  oxidation-and-reduction  reactions. 

A  metathetical  reaction  is  one  in  which  an  element  or 
group  of  elements  in  one  compound  changes  place  with 
some  other  element  or  group  in  another  compound.  All  the 
reactions  illustrated  by  equations  in  the  preceding  section 
are  metathetical  reactions,  except  the  combination  of  hydro- 
gen and  chlorine.  A  reaction  of  oxidation  and  reduction  is 
one  in  which  oxygen  or  some  acid-forming  element  or  group, 
or  hydrogen  or  some  base-forming  element  or  group,  is  taken 
away  from  one  compound  and  added  to  another  compound  ; 
the  compound  which  takes  up  oxygen  or  the  acid-forming 
element  or  loses  hydrogen  or  the  base-forming  element,  is 
said  to  be  oxidized,  and  the  compound  of  which  the  reverse 
is  true,  to  be  reduced.  The  following  equations  represent 
reactions  of  oxidation  and  reduction,  the  substance  which 
is  oxidized  being  placed  first  in  each  case. 

SnCl2  +  2HgCl2  =  SnCl4  +  2HgCl. 
2K4Fe(CN)6  +  C12  =  2K3Fe(CN)6  +  KG. 
3PbS  +  8HN03  =  3PbSO4  +  8NO  +  4H3O. 

If  now  we  recognize  that  in  correspondence  with  these 
two  important    classes  of  reactions  a  substance   may  have 


M  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

two  different  equivalent  weights,  and  if  we  adopt  a  definite 
quantity  of  some  substance  as  a  standard  of  reference,  the 
term  equivalent  weight  becomes  a  fairly  definite  one.  Since 
hydrogen  is  the  element  which  always  undergoes  replace- 
ment in  the  important  metathetical  reactions  in  which  acids 
or  bases  take  part,  and  since  it  is  also  frequently  involved  in 
reactions  of  oxidation  and  reduction,  it  is  most  natural  and 
•convenient  to  adopt  some  quantity  of  it  as  the  standard ; 
and  it  is  customary  to  select  as  this  quantity  its  combining 
weight  (approximately  1.0075  grams),  which  makes  the 
latter  identical  with  its  equivalent  weight,  and  makes  16 
grams  of  oxygen  the  ultimate  basis  of  the  system  of  equiva- 
lent weights  as  well  as  of  that  of  combining  weights.  The 
equivalent  weight  or  one  equivalent  of  a  substance  is  then 
defined  to  be  that  weight  of  it  which  enters  into  a  reaction 
of  simple  metathesis,  or  of  oxidation  and  reduction,  with  one 
equivalent  of  hydrogen  or  with  that  weight  of  any  other 
substance  which  itself  reacts  with  one  equivalent  of  hydro- 
gen. Whether  the  hydrogen  is  in  the  form  of  the  elemen- 
tary substance  or  in  that  of  one  of  its  compounds,  is  imma- 
terial. If  the  reaction  under  consideration  is  a  metathetical 
one,  the  quantity  just  defined  is  called  the  metathetical  equiv- 
alent, if  one  of  oxidation  and  reduction,  the  oxidation-equiv- 
alent, of  the  substance. 

The  following  examples  will  serve  to  illustrate  these 
.statements.  One  equivalent  of  any  acid  is  evidently  that 
weight  of  it  which  contains  one  equivalent  of  hydrogen 
-capable  of  reacting  with  a  base,  and  one  equivalent  of  any 
oase  is  that  weight  of  it  that  neutralizes  one  equivalent  of 
an  acid.  Thus,  the  formula- weights,  36.46  and  60.03  grams, 
of  hydrochloric  acid  (HC1)  and  of  acetic  acid  (HC2H3O2) 
are  also  the  equivalent  weights  of  these  substances,  for 
these  weights  contain  one  equivalent  (1.0075  grams)  of 
replaceable  hydrogen.  The  equivalent  weight  of  sulphu- 
ric acid  (H2SO4),  on  the  other  hand,  is  only  one-half  of  its 
formula-weight  (98.08  grams),  that  is,  49.04  grams.  Simi- 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.          66- 

larly,  the  equivalent  weight  of  sodium  hydroxide  (NaOH) 
is  identical  with  its  formula-weight,  while  that  of  barium 
hydroxide  (BaO2H2)  is  one-half,  and  that  of  ferric  hydrox- 
ide (FeO3H3)  one-third,  of  its  formula- weight.  The  equiv- 
alent weight  of  silver  nitrate  (AgNO3)  is  the  same  as  its 
formula-weight,  since  that  quantity  precipitates  one  equiva- 
lent of  hydrochloric  acid ;  those  of  barium  chloride  (BaCl2) 
and  ferric  chloride  (FeCl3)  are  one-half  and  one-third  of 
their  formula-weights  respectively,  since  it  is  these  quanti- 
ties that  are  produced  by  the  neutralization  of  one  equiva- 
lent of  hydrochloric  acid  by  the  corresponding  bases,  or 
since  they  react  with  one  equivalent  of  silver  nitrate. 

In  order  to  determine  the  oxidation-equivalent  of  a 
substance  it  is  necessary  to  know  into  what  other  substance 
it  is  reduced  or  oxidized,  and  to  consider  how  much  hydro- 
gen would  be  required  to  effect  its  reduction,  or  the  same 
reduction  of  other  substances  which  it  itself  is  capable  of 
effecting.  Thus  the  oxidation-equivalent  of  ferric  chloride 
with  respect  to  its  reduction  to  ferrous  chloride  (FeCl2)  is 
the  weight  corresponding  to  the  formula  FeCl3  ;  that  of 
stannous  chloride  with  respect  to  its  oxidation  to  stannic 
chloride  (SnCl4)  is  that  corresponding  to  the  formula  £SnCl2, 
and  those  of  manganese  heptoxide  (Mn2O7)  with  respect  to 
its  reduction  to  MnO2  and  to  MnO  are  the  weights  corre- 
sponding to  iMn2O7  and  to  J^Mn2O7,  respectively.  The 
equivalent  weights  of  elementary  substances  are  similarly 
determined;  thus,  it  follows  from  the  formulas  AgCl,  ZnCl2, 
and  A1C13,  and  the  combining  weights  of  the  elements  (§  16), 
that  the  equivalent  weight  of  silver  is  107.93,  that  of  zinc, 
32.70,  and  that  of  aluminum,  9.037.  In  cases  where  an 
element  exists  in  two  different  states  of  oxidation,  like 
mercury  in  mercurous  and  mercuric  chloride  (HgCl  and 
HgCl2),  or  tin  in  stannous  and  stannic  chloride  (SnCl2  and 
SnCl4),  it  has  two  different  equivalent  weights  with  refer- 
ence to  its  conversion  into  each  of  these  states ;  thus  the 
two  values  are  200  and  100  for  mercury,  and  59.50  and 


56          GENERAL   PRINCIPLES  OF  PHYSICAL   SCIENCE. 

29.75  f°r  tm»  since  the  combining  weights  of  these  ele- 
ments are  200.0  and  119.0,  respectively.  These  examples 
and  that  of  manganese  hept oxide  just  preceding  them 
illustrate  the  necessity,  in  cases  where  a  substance  can  be 
converted  into  two  or  more  other  states  of  oxidation,  of 
specifying  in  connection  with  the  equivalent  weight  what 
reaction  is  under  consideration. 

An  important  unit  of  concentration,  which  is  most  com- 
monly used  in  connection  with  solutions,  is  also  based  on  the 
definition  of  equivalents  ;  namely,  when  one  equivalent  of  a  sub- 
stance is  contained  in  one  liter  of  a  solution,  the  concentration 
of  the  substance  is  said  to  be  normal,  and,  in  general,  concen- 
tration values  are  expressed  by  prefixing  to  the  word  normal 
a  number  equal  to  that  of  the  equivalents  contained  in  one 
liter.  Thus,  a  o.i  normal  solution  of  sulphuric  acid  is  one 
containing  4.904  grams  of  it  in  a  liter  of  solution.  Designat- 
ing the  number  of  equivalents  of  any  one  substance  by  N 
(which  is  equal  to  m  /  A,  where  m  is  the  weight  and  A  the 
equivalent  weight  of  the  substance),  and  designating  by  v  the 
volume  of  the  solution  in  which  they  are  contained,  the  equiv- 
alent concentration  c  is  defined  by  the  relation,  c  =  N  /  v. 

20.  General  Significance  of  the  Properties  of  Gases.  — 
The  properties  of  gases  under  moderate  pressure  are  of  so 
great  importance,  not  only  from  the  stand-point  of  general 
chemistry,  but  also  from  that  of  the  laws  of  energy,  that  they 
deserve  consideration  among  the  general  principles  of  physical 
science.     The  laws  regarding  the  relation  of  their  volume 
to  pressure,  temperature,  and  combining  weight  will  be  here 
discussed,  while  those  regarding  their  energy-relations  will  be 
considered  in  the  next  Chapter. 

21.  Relation  between  the  Pressure  and  Volume  of 
Gases.      Boyle's   Law.  —  In    the    case    of    gases    having 
pressures  not  greatly  exceeding  that  of  the  atmosphere,  the 
following   simple   relation,   known    from    its    discoverers   as 
Boyle  s  or  Mariotte  s  Law,  has  been  found  to  exist  between 
pressure  and  volume :     At  constant  temperature,  the  volume 


GENERAL  PRINCIPLES  RELATING  TO  MATTER.          57 

of  a  definite  quantity  of  gas  is  inversely  proportional  to  its 
pressure.  Since  the  density  of  a  substance  is  defined  to 
be  the  ratio  of  its  mass  to  its  volume,  this  law  may  also  be 
stated  as  follows  :  At  constant  temperature,  the  density  of  a 
gas  is  directly  proportional  to  its  pressure.  Thus,  if  a 
quantity  of  gas  occupied  a  volume  of  one  liter  at  a  pressure 
of  one  atmosphere,  its  volume  at  a  pressure  of  two  atmos- 
pheres would  be  half  a  liter ;  at  three  atmospheres,  one-third 
of  a  liter ;  and  so  forth :  and  its  density,  or  the  weight  of 
i  ccm.  of  it,  would  be  twice  as  great  at  two  atmospheres, 
three  times  as  great  at  three  atmospheres,  and  so  forth. 
Boyle's  Law  is  expressed  mathematically  as  follows : 

i>  v        i> 

£_  =  *L  =  const., 
m        D 

where  /,  v,  m,  and  D  are  the  pressure,  volume,  mass,  and 
density,  respectively,  of  a  definite  kind  of  gas  at  a  definite 
temperature. 

The  units  employed  for  the  measurement  of  mass  and 
volume  have  been  already  defined,  but  a  statement  is 
necessary  in  regard  to  those  used  for  the  measurement  of 
pressure.  Pressure  is  expressed  in  the  centimeter-gram- 
second  system  in  dynes  (a  unit  defined  in  §  29)  per  square 
centimeter.  Another  unit  of  pressure  that  is  frequently  em- 
ployed is  the  atmosphere,  which  is  defined  to  be  equal  to  the 
pressure  exerted  by  a  column  of  mercury,  76  cm.  in  height, 
of  a  density  of  13.59593  (which  is  almost  exactly  the  value 
at  o°),  and  submitted  to  the  normal  intensity  of  gravity, 
which  is  so  defined  as  to  be  almost  exactly  equal  to  the 
average  intensity  at  the  sea-level  in  the  latitude  of  45°.  One 
atmosphere  is  approximately  the  mean  value  of  the  atmos- 
pheric pressure  at  the  sea-level.  It  is  also  very  common  in 
experimental  work  to  express  pressure  in  terms  of  the  height 
of  a  mercury  column,  reduced  to  the  conditions  just  described, 
which  exerts  a  pressure  equal  to  that  to  be  measured. 

There  is  scarcely  any  doubt  that  Boyle's  Law  is  also 
applicable  to  the  components  of  a  mixture  of  two  or  more 


58          GENERAL   PRINCIPLES  OF  PHYSICAL   SCIENCE. 

gases.  That  is,  each  component  of  a  gaseous  mixture  exerts 
the  same  pressure  as  it  would  if  it  were  alone  present  in 
the  volume  occupied  by  the  mixture.  The  pressures  exerted 
by  the  separate  components  are  called  the  partial  pressures, 
and  this  principle  in  regard  to  their  values  is  known  as 
Daltons  Law.  The  law  has  not  received  direct  experimental 
verification,  owing  to  the  difficulty  of  obtaining  satisfactory 
semipermeable  walls  (§  8)  against  which  the  partial  pressures 
might  be  accurately  measured ;  but  its  validity  is  made  almost 
certain  by  the  fact  that  the  total  pressure  exerted  by  a 
gaseous  mixture,  which  is  of  course  the  sum  of  the  partial 
pressures,  is  found  to  be  equal  to  the  sum  of  the  pressures 
calculated  by  Boyle's  Law  for  the  separate  components.  In 
illustration  of  this  law,  the  following  facts  may  be  cited. 
When  two  liters  of  oxygen  and  one  liter  of  nitrogen,  both  at 
the  pressure  of  one  atmosphere,  are  mixed,  and  the  volume 
of  the  mixture  is  maintained  at  three  liters,  the  pressure  is 
found  to  be  still  one  atmosphere,  which  is  in  accordance  with 
Dalton's  Law ;  for  the  partial  pressure  of  the  oxygen  should 
have  become  two-thirds,  and  that  of  the  nitrogen,  one-third 
of  an  atmosphere.  If  the  volume  of  the  mixture  is  reduced 
to  one  liter,  the  pressure  is  found  to  be  three  atmospheres, 
the  partial  pressures  of  the  oxygen  and  nitrogen  having 
undoubtedly  become  two  atmospheres  and  one  atmosphere, 
respectively,  as  Dalton's  Law  requires. 

A  gas  which  is  considered  to  conform  completely  to 
Boyle's  Law  is  called  a  perfect  or  ideal  gas.  No  such  gas 
actually  exists,  however  ;  for  Boyle's  Law  is  not  in  any  case 
absolutely  exact.  In  the  case  of  gases  whose  temperature 
is  far  above  that  of  liquefaction,  the  deviations  from  it  are 
small  until  high  pressures  are  reached ;  in  the  case  of  gases 
in  the  neighborhood  of  their  points  of  liquefaction,  consider- 
able deviations  exist,  even  at  low  pressures.  The  following 
table  gives  an  idea  of  the  magnitude  and  direction  of  the 
deviations  in  the  case  of  some  common  representative  gases. 
The  values  in  the  columns  headed  p\v\  :  /2^2  are  the  ratios 


GENERAL   PRINCIPLES  KELA  TING   TO  MA  TTER. 


59 


of  the  pressure-volume  product  at  one  atmosphere's  pressure 
(pi)  to  that  at  two  atmospheres'  pressure  (/a),  at  a  temper- 
ature of  8°,  unless  otherwise  specified.  Between  these  pres- 
sures, the  deviations  will  be  seen  to  vary  from  about  0.05  to  2.6 
per  cent,  in  the  case  of  the  different  gases.  The  table  also 
contains  some  examples  illustrating  the  effect  of  temperature 
on  the  deviations. 


Name  of  the  gas. 

A»i  :  /2»j- 

Name  of  the  gas. 

/i»i  :  A»»- 

Hydrogen 

0.9995 
1.0007 
1.0028 
1.0065 
1.0188 
1.0235 

(8° 
Carbon  dioxide    }  50° 
(200° 
(8° 
Sulphur  dioxide  {  50° 
(200° 

1.0065 
1.0036 
1.0008 
1.0209 
1.0110 
1.0021 

Nitric  oxide  .... 
Nitrous  oxide  .  .  . 
Ammonia  • 

As  the  pressure  continues  to  increase,  in  all  cases  except 
that  of  hydrogen,  the  values  of  the  product  pv  continue  to 
decrease,  pass  through  a  minimum,  and  then  steadily  increase ; 
in  the  case  of  hydrogen,  a  regular  increase  takes  place  from 
the  start. 

Expressed  in  mathematical  form,  it  is  found  that  the 
behavior  of  hydrogen  up  to  very  high  pressures  is  repre- 
sented by  the  equation,  /  (v  —  b)  =  const.,  and  that  of  other 
gases  by  the  so-called  van  der  Waals*  equation, 


P  +  (v—b  )  =const., 


in  which  equations  a,  by  and  const,  represent  constants,  which 
vary  with  the  nature  and  quantity  of  the  gas.  For  example, 
in  the  case  of  ethylene,  a  gas  which  deviates  greatly  from 
Boyle's  Law,  the  values  at  20°  of  p  v  calculated  with  the  help 
of  this  last  equation  agree  closely  with  those  experimentally 
determined,  even  up  to  a  pressure  of  400  atmospheres,  if 
appropriate  values  of  the  constants  be  assumed.  This  is 
shown  by  the  following  table,  in  which  are  given  the  pressures, 


60 


GENERAL   PRINCIPLES   OF  PHYSICAL  SCIENCE. 


expressed  in  atmospheres,  and  the  observed  and  calculated 
values  of  /  v,  an  arbitrary  unit  of  volume  being  employed. 


Pressures 

31.6 

46 

84 

133 

234 

329 

399 

p  v  observed 

0.914 

0.781 

0.399 

0.520 

0.807 

1.067 

1.248 

p  v  calculated 

0.895 

0.782 

0.392 

0.520 

0.805 

1.067 

1.254 

The  following  figure  illustrates  the  behavior  of  hydro- 
gen and  nitrogen  at  60°,  of  carbon  dioxide  at  two  temper- 
atures (60°  and  40°),  and  of  a  perfect  gas  (marked  P.  G.  in 
the  figure),  up  to  a  pressure  of  320  meters  of  mercury. 
The  values  of  the  product  /  v  are  plotted  as  ordinates,  and 
the  pressures  (in  meters  of  mercury)  as  abscissas ;  such 
quantities  of  the  gases  being  taken  that  the  initial  value  of 
pv  is  the  same  in  each  case. 


V 


\ 


120  l6o  200  340 

PRESSURES. 


28O 


32° 


GENERAL   PRINCIPLES  RELATING   TO  MATTER.          61 

22.  Relation  between  the  Pressure- Volume  Product 
and  Temperature.  Gay-Lussac's  Law  of  Temperature- 
Effect. —  The  following  principle,  known  as  Gay-Lussac's 
Law  of  Temperature-Effect,  has  also  been  experimentally 
established :  A  definite  change  of  temperature  causes,  in 
the  case  of  different  gases,  the  same  fractional  change  in  the 
pressure-volume  product :  if  the  pressure  is  kept  constant, 
it  causes  in  all  gases  the  same  fractional  change  in  volume ; 
and  if  the  volume  is  kept  constant,  it  causes  in  all  gases  the 
same  fractional  change  in  pressure. 

For  example,  when  1000  ccm.  of  air  at  o°  are  heated  at 
the  atmospheric  pressure  to  100°,  the  volume  becomes  1367 
ccm. ;  and  when  500  ccm.  of  hydrogen  are  heated  in  the 
same  manner,  the  volume  becomes  about  683.5  ccm.;  the 
fractional  increase  (36.7  per  cent.)  being  the  same  in  the  two 
cases.  If  a  quantity  of  either  air  or  hydrogen,  having  at  o° 
a  pressure  of  one  atmosphere,  is  heated  to  100°  in  a  vessel  of 
constant  volume,  the  pressure  increases  to  1.367  atmospheres. 

It  will  be  noted  that  this  law  states  only  that  temper- 
ature has  the  same  effect  in  the  case  of  different  gases,  but 
does  not  show  what  the  functional  relation  is  between 
temperature  and  the  pressure-volume  product.  This  relation 
will,  of  course,  depend  on  the  scale  that  has  been  adopted  for 
the  measurement  of  temperature ;  specifications  in  regard  to 
this  scale  must  therefore  be  here  presented.  It  has  been 
agreed  by  physicists  to  adopt  as  the  unit  for  the  measure- 
ment of  temperature  the  centigrade  degree,  which  is  defined 
to  be  the  difference  in  temperature  which  produces  in  the 
pressure  of  a  quantity  of  hydrogen  gas  one  one-hundredth 
part  of  the  variation  in  its  pressure  which  occurs  when  its 
volume  is  kept  constant  and  its  temperature  is  changed 
from  that  of  ice  melting  to  that  of  water  boiling  under  the 
pressure  of  one  atmosphere.  As  it  has  been  found  that 
the  fractional  increase  in  pressure  varies  slightly  with  the 
initial  pressure  of  the  gas,  it  is  necessary  to  specify  also  a 
definite  initial  pressure ;  accordingly,  it  is  agreed  that  the 


62  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

pressure  of  the  gas  at  the  temperature  of  melting  ice  shall 
be  that  of  one  meter  of  mercury.  In  order  to  express  the 
temperature  absolutely,  it  is  also  necessary  to  adopt  some 
arbitrary  zero-point  ;  and  as  such,  the  temperature  of  ice 
melting  under  a  pressure  of  one  atmosphere  has  been  chosen. 
Temperatures  referred  to  this  scale  and  to  this  zero-point 
are  designated  normal  temperatures  (t).  In  many  theo- 
retical considerations  it  is  simpler  to  adopt  such  a  zero-point 
that  temperatures  will  be  directly  proportional  to  the 
pressures  which  they  produce  in  a  quantity  of  hydrogen 
gas  ;  now,  since  its  pressure  at  the  temperature  of  melting 
ice  is  found  to  increase  ^70  when  the  gas  is  heated  to  100°, 
this  proportionality  will  be  secured  if  a  zero-point  273.0° 
lower  than  the  normal  zero  be  adopted.  Temperatures 
referred  to  this  zero-point  are  called  absolute  temperatures 
(T),  by  reason  of  a  relation  to  the  Second  Law  of  Ener- 
getics that  will  be  explained  in  Chapter  IV.  It  is  evident 
that  T=  273.0  +  /. 

Now,  since  the  change  in  pressure  of  hydrogen  gas 
has  been  adopted  as  the  ultimate  standard  for  the  measure- 
ment of  temperature,  and  since  by  the  Law  of  Gay-Lussac 
temperature  has  the  same  effect  in  the  case  of  all  gases,  it 
follows  that  the  change  in  the  pressure-volume  product  of  any 
gas  is  directly  proportional  to  its  change  in  temperature. 
This  statement,  which  embraces  the  definition  of  temper- 
ature and  the  Laws  of  Gay-Lussac  and  Boyle,  is  mathe- 
matically expressed  by  the  equation  : 

pv  =  pQvo(i  +  ex/), 


where  <x  is  a  quantity  (known  as  the  coefficient  of  expansion) 
which  has  nearly  the  same  value  for  all  gases,  and  where 
/o>  ^o,  and  /,  v,  represent  the  corresponding  pressures  and 
volumes  of  the  gas  at  o°  and  /°,  respectively.  The  value  of  «, 
which  is  by  definition  equal  to  the  fractional  increase  which 
the  pressure-volume  product  undergoes  when  a  gas  is  heated 
from  o°  to  i°,  has  been  already  stated  to  be  ^^  in  the 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.          63 

case  of  hydrogen.     By  substituting  this  value,  the  last  equa- 
tion can  be  transformed  into  the  following  one  : 


pv 

=  * ""  =  m  X  const,  (for  a  definite  gas). 


/  +  273  ~  '   273 

With  the  help  of  the  concept  of  absolute  temperature, 
this  principle  may  be  stated  more  simply  as  follows :  The 
pressure-volume  product  of  a  definite  quantity  of  any  definite 
gas  is  proportional  to  its  absolute  temperature.  It  is  ex- 
pressed by  the  equation : 

pv 

—  =  m  X  const., 

in  which  the  constant  is  evidently  the  value  of  the  ratio 
/  v  I  T  for  i  gram  of  any  definite  gas.  Its  value  is  different 
for  different  gases,  however. 

Attention  may  also  be  called  to  the  fact,  which  is  some- 
times overlooked,  that,  though  the  absolute  increase  in  the 
pressure  or  volume  of  a  definite  quantity  of  a  gas  is  the  same 
for  i°,  whatever  may  be  the  temperature,  the  relative  or 
fractional  increase  is  variable,  being  always  one  7th  part  of 
the  initial  value.  For  example,  a  gas  increases  ^-g  of  its 
volume  measured  at  o°  whether  it  is  heated  at  constant 
pressure  from  o°  to  i°  or  from  100°  to  101°;  but  in  either 
case  it  increases  not  ^^,  but  ^^,  of  its  volume  measured  at 
1 00°.  The  ratio  pv  I  T  is  evidently  the  absolute  increase 
in  the  product  pv  due  to  an  increase  of  temperature  of  i° 
at  any  temperature  T. 

The  Law  of  Gay-Lussac,  like  that  of  Boyle,  is  not  abso- 
lutely exact ;  that  is,  equal  changes  in  temperature  do  not 
produce  in  all  gases  exactly  the  same  change  in  the  product 
pv.  This  is  shown  by  the  following  table,  which  contains 
the  values  of  the  fractional  increase  in  pressure  (100  «)  which 
different  gases  undergo  when  they  are  heated  from  o°  to 
1 00°,  the  volume  being  kept  constant  and  the  initial  pres- 
sure being  that  of  one  meter  of  mercury. 


64 


GENERAL    PRINCIPLES  OF  PHYSICAL  SCIENCE. 


Name  of  the  gas. 

100*. 

Name  of  the  gas. 

IOOOC. 

Hydroffen       .... 

03662 

Carbon  dioxide 

0  3725 

0.3675 

Sulphur  dioxide  . 

03845 

Carbon  monoxide    .     . 

0.3667 

It  will  be  seen  that  all  the  very  difficultly  liquefiable 
gases  (those  contained  in  the  first  column)  have  nearly  the 
same  values,  while  the  more  easily  liquefiable  gases  (those  con- 
tained in  the  second  column)  have  considerably  larger  values. 

23.  Relation  between  the  Pressure-Volume  Product 
and  Combining  Weight.  Gay-Lussac's  Law  of  Combin- 
ing Volumes.  —  In  addition  to  the  laws  already  considered, 
an  important  relation  between  the  pressure-volume  product 
for  substances  in  the  gaseous  condition  and  their  combining 
weights  has  been  established  by  experiment.  This  relation 
may  be  stated  as  follows  :  The  combining  weights  of  all  sub- 
stances in  the  gaseous  condition  have  at  the  same  temperature 
equal  values  of  the  pressure-volume  product,  or  values  of  it 
which  stand  to  one  another  in  the  ratio  of  small  whole  num- 
bers. If  the  volume  occupied  by  the  combining  weight  of 
a  substance  is  called  its  combining  volume,  the  law  may  also 
be  stated  thus :  The  combining  volumes,  measured  at  the 
same  temperature  and  pressure,  of  all  substances  in  the  gas- 
eous condition  are  equal,  or  stand  to  one  another  in  the  ratio 
of  small  whole  numbers.  This  law  is  called  the  Gay- 
Lussac's  Law  of  Combining  Volumes. 

For  example,  two  volumes  of  hydrogen  unite  with  one 
volume  of  oxygen  to  form  water ;  one  volume  of  ammonia 
unites  with  one  volume  of  hydrochloric  acid  gas  to  form 
ammonium  chloride ;  two  volumes  of  benzene-vapor  require 
fifteen  volumes  of  oxygen  for  their  complete  combustion, 
and  twelve  volumes  of  carbon  dioxide  and  six  volumes  of 
water-vapor  are  formed  as  products.  To  substances  which 
are  not  together  involved  in  any  one  definite  chemical  reac- 
tion the  principle  is  also  applicable;  for  their  combining 


GENERAL  PRINCIPLES  RELATING   TO  MATTER.          65 

weights  can  always  be  determined  indirectly,  by  comparisons 
of  their  elementary  compositions  or  of  the  quantities  of  them 
involved  in  reactions  with  other  substances.  Thus,  the  com- 
bining weights  of  ammonia  and  hydrogen,  determined,  for 
example,  by  comparing  the  quantities  of  them  combined  with 
a  definite  quantity  of  chlorine  in  the  compounds  ammo- 
nium chloride  and  hydrochloric  acid,  are  to  each  other  as 
17.06  :  1.008,  and  the  combining  volumes  of  these  sub- 
stances, that  is,  the  volumes  of  17.06  grams  of  ammonia 
and  of  i  .008  grams  of  hydrogen,  are  to  each  other  as  2  :  i . 
Gay-Lussac's  Law  of  Combining  Volumes  is  subject  to 
deviations  of  the  same  order  of  magnitude  as  those  which 
affect  the  validity  of  the  Boyle's  Law  and  Gay-Lussac's  Law 
of  Temperature-Effect.  For  example,  the  volumes  of  hydro- 
gen and  oxygen,  measured  at  o°  and  76  cm.  pressure,  which 
combine  with  each  other  to  form  water  are  not  as  2  :  i,  but, 
according  to  the  accurate  experiments  of  Scott  and  Morley, 
as  2.0027  :  i.  In  the  case  of  readily  liquefiable  gases,  the 
deviations  from  whole  numbers  are  much  greater. 

24.  General  Expression  of  the  Pressure-Volume 
Relations  of  Gases.  Empirical  Definition  of  Molecular 
Weight.  —  It  was  mentioned  above  that  the  product  pv I  T 
has  a  different  value  for  equal  weights  of  different  gases ; 
thus,  when  absolute  units  are  employed,  it  has  the  value  of 
2  600000  for  i  gram  of  oxygen,  and  the  value  41  200000 
for  i  gram  of  hydrogen.  Since,  however,  its  value  for  dif- 
ferent weights  of  a  definite  gas  is  proportional  to  those 
weights,  it  is  evident  that  the  relative  weights  of  different 
gases  which  have  the  same  value  of  the  product  pv  I  T  can 
be  readily  calculated,  and  that  these  weights  can  be  expressed 
absolutely  by  adopting  a  definite  weight  of  some  definite  gas 
as  a  standard  of  reference.  It  has  been  agreed  to  adopt 
32  grams  of  oxygen  as  the  standard,  and  to  designate  that 
quantity  of  any  other  gas  that  has  the  same  value  of  the 
product/^  /  T  as  32  grams  of  oxygen,  the  molecular  weight, 
or  one  mo  I,  of  the  gas. 


66  GENERAL   PRINCIPLES  OF  PHYSICAL  SCIENCE. 

In  consequence  of  the  Law  of  Combining  Volumes  and 
the  adoption  of  twice  the  combining  weight  of  oxygen  as  the 
standard,  it  is  evident  that  the  molecular  weight  of  any  sub- 
stance is  equal  to  its  combining  weight,  or  stands  to  it  in  the 
relation  of  some  small  whole  number.  But  since  the  Law  of 
Combining  Volumes  is  not  exact,  and  since  as  a  rule  gaseous 
densities  are  not  determined  with  as  great  accuracy  as  the 
combining  weights,  the  following,  exact  definition,  based  both 
on  combining  weights  and  on  the  volume-relations  of  gases, 
has  been  adopted  :  the  molecular  weight  or  one  mol  of  any 
gaseous  substance  is  that  small  multiple  or  sub-multiple  of 
its  combining  weight  which  has  approximately  the  same  value 
of  the  product  /  v  /  T  as  32  grams  of  oxygen. 

While  the  name  molecular  weight  originated  from  cer- 
tain hypothetical  considerations  connected  with  the  quantities 
denoted  by  it,  it  is  clear  that  these  quantities  as  here  defined 
are  experimentally  determinable  ones,  and  are  not  based  on 
any  hypothesis  whatever. 

With  the  help  of  this  definition  of  molecular  weight,  the 
volume-relations  of  gases  under  moderate  pressures  can  be 
fully  expressed  by  the  following  highly  important,  general 
equation  : 


in  which  /,  v,  and  T  signify,  as  usual,  the  pressure,  volume, 
and  absolute  temperature  of  the  gas  ;  TV  is  the  number  of  mols 
present  in  it,  and  is  evidently  equal  to  m  /  M,  where  m  is  the 
weight,  and  M  the  molecular  weight,  of  the  gas  ;  and  R  is  a 
constant  for  all  gases,  equal  to  the  value  of  /  v  /  T  for  one 
mol  of  any  gas. 

The  numerical  value  of  the  constant  R  is  readily  calcu- 
lated from  the  data  pertaining  to  oxygen  :  according  to  the 
exact  determinations  of  Morley,  one  gram  of  this  gas  under 
the  so-called  normal  conditions,  that  is,  at  o°  and  a  pressure 
of  one  atmosphere,  occupies  a  volume  of  699.8,  or  almost 
exactly  700  ccm.,  so  that  32  grams  of  it  occupy  a  volume  of 
22  400  ccm.  or  22.4  liters,  a  value  which  (like  all  others  that 


GENERAL   PRINCIPLES  RELATING    TO  MATTER.        67 

in  this  book  are  represented  by  bold  type)  it  is  well  to  remem- 
ber, since  it  is  the  volume  of  one  mol  of  any  gas  under  normal 
conditions.  Therefore, 

*  =  /l='X   22.40 

NT      i  X  273.0 

when  the  pressure  is  expressed  in  atmospheres  and  the  vol- 
ume in  liters.  If  the  pressure  is  expressed  in  dynes  and 
the  volume  in  cubic  centimeters,  R  =  8.31  X  io7. 

It  is  evident,  if  four  of  the  five  factors  (m,  M,p,  v,  T) 
on  which  depend  the  volume  properties  of  gases,  are  known, 
that  the  remaining  one  can  readily  be  calculated  by  this 
equation.  As  in  all  numerical  applications  of  algebraic  ex- 
pressions, care  must  be  taken  that  the  units  in  which  the 
corresponding  quantities  (here  the  constant  R  and  the  five 
variables)  are  expressed,  be  identical. 

By  dividing  both  members  of  the  equation  /  v  =  NR  T  by 
m,  the  weight  of  the  gas,  and  substituting  for  m  /  v  the  den- 
sity D,  and  form/N  the  molecular  weight  M  of  the  gas,  the 
equation,  M=  DR  T I  /,  is  obtained.  This  states  that,  at 
constant  temperature  and  pressure,  the  molecular  weights 
of  gases  are  proportional  to  their  (absolute)  densities.  Since 
the  molecular  weight  of  oxygen  Mo  =  32,  the  molecular 
weight  of  any  gas  is  equal  to  32  times  the  ratio  of  its 
density  to  that  (Do)  of  oxygen  at  the  same  temperature  and 
pressure ;  that  is,  M  =  32  D  /  Do .  Since  experimentally  de- 
termined densities  are  often  referred  to  the  density  of  air 
(Z?A)  instead  of  that  of  oxygen,  it  is  worthy  of  note  that 
£>A  :  D0  :  :  28.97  •  32»  or  very  nearly  as  29  :  32. 

It  is  customary  to  employ  for  gaseous  substances  whose 
densities  have  been  determined,  chemical  formulas  corre- 
sponding to  that  multiple  of  the  combining  weight,  which 
is  the  molecular  weight  as  above  defined.  Such  formulas 
are  called  molecular  formidas.  They  evidently  express  not 
only  the  nature  and  relative  quantities  of  the  component 
elements,  but  also  the  value  of  the  pressure-volume  product 
for  any  weight  of  the  gaseous  substance.  The  formula- 


68  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

weight  of  a  gaseous  substance  thus  becomes  identical  with 
its  molecular  weight.  It  is  also  the  practice,  when  reasons 
for  another  procedure  do  not  exist,  to  represent  the  liquid 
and  solid  substances  into  which  gaseous  substances  are 
converted  by  withdrawal  of  heat,  by  the  same  molecular 
formulas,  and  to  designate  the  weights  corresponding  to 
these  formulas  as  one  mol  of  the  substances. 

For  example,  liquid  benzene  is  found  by  analysis  to 
contain  92.25  per  cent,  of  carbon  and  7.75  per  cent,  of 
hydrogen.  By  dividing  these  percentages  by  the  combining 
weights  of  the  elements  (12.00  and  1.008,  respectively),  it 
is  found  that  the  substance  contains  an  equal  number  of 
combining  weights  of  the  two  elements.  Its  combining 
weight  is  therefore  13.01  and  its  empirical  formula  is  CH. 
Its  density,  when  in  the  form  of  vapor  at  100°,  has  been 
determined  and  found  to  be  2.47  times  as  great  as  that  of 
oxygen  at  that  temperature  when  under  the  same  pressure. 
The  molecular  weight  of  it  is  therefore  approximately 
32  X  2.47  =  79.0,  which  is  nearly  six  times  the  combining 
weight  deduced  from  the  analysis.  The  exact  molecular  weight 
is  therefore  78.06,  and  the  molecular  formula  is  C6H6. 

It  is  frequently  desirable  to  express  the  concentration  of 
a  substance  in*  terms  of  the  number  of  mols  per  unit  of 
volume ;  and  it  has  recently  been  proposed  to  use  the 
adjective  molar  to  signify  one  mol  per  liter,  and  to  prefix 
to  this  word  numbers  to  indicate  fractions  and  multiples  of 
one  mol  per  liter.  Thus,  a  0.05  molar  solution  of  benzene  in 
alcohol  is  one  which  contains  -fa  mol  or  3.903  grams  of  ben- 
zene in  one  liter  of  the  solution.  In  general,  designating 
by  N  the  number  of  mols  of  any  one  substance  contained 
in  the  volume  vt  the  molar  concentration  C  is  expressed  by 
the  relation,  C  =  N I  v.  The  general  adoption  of  this  term 
molar  seems  highly  desirable,  in  order  to  avoid  the  serious 
confusion  which  is  beginning  to  arise  through  the  use  of  the 
term  normal  by  some  writers  in  the  sense  of  one  mol,  as  well 
as  in  its  appropriate  sense  of  one  equivalent,  per  liter. 


CHAPTER  IV. 

THE    GENERAL    PRINCIPLES     RELATING    TO     ENERGY. 

25.  The  Forms  of  Energy  and  Other  Classes  of 
Energy  Manifestations.  —  Energy  has  been  already  denned 
(§  6)  to  be  that  which  gives  rise  to  the  changes  in  the  prop- 
erties of  bodies  and  to  the  power  which  bodies,  or  in  some 
cases  portions  of  unoccupied  space,  possess  of  producing 
such  changes.  It  has  also  been  stated  that  the  manifesta- 
tions of  energy  are  most  varied  in  character.  Nevertheless, 
it  has  been  found  possible  to  refer  all  of  them  to  a  compara- 
tively small  number  of  so-called  forms  of  energy,  each  form 
corresponding  to  a  certain  definite  tendency  in  the  body 
possessing  it  to  undergo  a  change  in  position  or  condition. 
The  forms  of  energy  which  may  be  associated  with  matter 
are  designated  as  follows: 

1.  Kinetic  Energy.  5.  Electrical  Energy. 

2.  Gravitation  Energy.  6.  Magnetic  Energy. 

3.  Cohesion  Energy.  7.  Chemical  Energy. 

4.  Disgregation  Energy.  8.  Heat  Energy. 

Kinetic  energy  is  the  energy  that  bodies  possess  in 
virtue  of  their  motion.  Gravitation  energy  is  the  energy 
that  bodies  possess  in  virtue  of  their  inherent  tendency  to 
approach  one  another.  Cohesion  energy  and  disgregation 
energy  are  terms  here  employed  to  designate  the  forms  of 
energy  a  body  possesses  in  virtue  of  the  tendency  of  its 
particles  to  approach  one  another,  and  to  recede  from  one 
another,  respectively.  The  existence  of  cohesion  energy  is 
shown  by  the  contraction  of  solid  and  liquid  bodies  upon 
cooling,  by  the  tendency  of  a  stretched  piece  of  wire  or 
rubber  to  become  shorter,  by  the  attraction  of  two  carefully 
ground  plates  laid  close  together,  and  by  the  tendency  of 
the  surface  of  liquids  to  diminish  in  extent,  as  is  illustrated 

69 


70  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

by  drops  of  water  assuming  a  spherical  form  and  by  capillary 
phenomena,  such  as  the  rise  of  liquids  in  narrow  tubes. 
Disgregation  energy  is  exhibited  by  all  gases  and  by  com- 
pressed liquids  and  solids  ;  for  these  have  the  power  of  pro- 
ducing changes  in  virtue  of  the  tendency  of  their  particles 
to  separate.  Electrical  and  magnetic  energies  cannot  be 
intelligibly  defined  in  a  few  words ;  it  is  sufficient  at  this 
point  to  state  that  they  are  the  forms  of  energy  correspond- 
ing to  those  properties  of  bodies  which  are  treated  of  in  the 
divisions  of  physics  known  as  Electricity  and  Magnetism. 
Chemical  energy  is  the  energy  that  bodies  possess  in  virtue 
of  the  tendency  of  the  chemical  substances  present  in  them 
to  undergo  transformations  into  other  substances.  Heat 
energy  is  the  energy  that  bodies  possess  in  virtue  of  the 
tendency  of  their  temperatures  to  decrease  :  it  is  the  energy 
which  is  given  out  by  bodies  when  they  are  brought  into 
communication  with  surroundings  of  lower  temperature. 

Energy  manifestations  may  also  be  considered  from 
other  points  of  view.  A  convenient  classification  of  some 
of  them,  based  on  the  space  relations  of  bodies,  may  be 
here  mentioned.  The  energy  which  bodies  possess  in  virtue 
of  their  tendency  to  approach  or  recede  from  one  another  is 
called  distance  energy,  without  reference  to  the  form  of 
energy,  as  above  defined,  from  which  the  tendency  may 
arise ;  thus,  gravitation  energy  is  by  nature  a  form  of  dis- 
tance energy,  but  electrical  and  magnetic  energies  also  give 
rise  to  it  under  certain  conditions,  as  is  illustrated  by  the 
attraction  and  repulsion  which  electrically  charged  bodies  or 
magnetized  bodies  exert  on  one  another.  The  powers  of 
doing  work  (a  term  defined  just  below)  which  bodies  possess 
in  virtue  of  their  tendencies  to  undergo  changes  in  surface, 
volume,  and  form,  are  called  surface  energy,  volume  energy, 
and  elastic  energy,  respectively.  These  are  special  mani- 
festations of  cohesion  and  disgregation  energies  rather 
than  distinct  forms  of  energy.  Surface  energy  is  of 
especial  importance  in  the  case  of  liquid  bodies,  whose 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.          71 

surfaces  always  tend  to  diminish  in  extent,  as  was  illustrated 
above.  Volume  energy  is  exhibited  by  all  compressed  bodies, 
whether  solid,  liquid,  or  gaseous.  Elastic  energy,  as  distinct 
from  volume  energy,  is  possessed  only  by  solid  bodies :  it  is 
the  energy  that  is  developed  in  them  by  the  external  appli- 
cation of  pressure  or  tension,  and  it  is  the  resultant  of  the 
changes  thereby  produced  in  the  cohesion  and  disgregation 
energies  within  the  bodies.  Thus,  the  elastic  energy  of  a 
bent  steel  spring  or  a  stretched  wire  is  due  both  to  the 
cohesion  energy  of  the  extended  parts  and  to  the  disgrega- 
tion energy  of  the  compressed  parts  of  the  spring  or  wire. 
In  the  interpretation  of  most  phenomena  the  consideration 
of  the  surface,  volume,  and  elastic  energies  has  a  great 
advantage  over  the  consideration  of  the  cohesion  and  dis- 
gregation energies  involved,  since  the  former  are  determined 
by  their  external  manifestations  and  are  therefore  capable  of 
quantitative  measurement  (as  is  described  in  §  30),  while, 
as  a  rule,  the  latter  are  not  accessible  to  it. 

The  general  term  potential  energy  is  used  to  designate 
the  energy  that  bodies  possess  in  virtue  of  their  position  or 
their  configuration:  it  includes  the  four  classes  of  energy- 
manifestations  considered  in  the  preceding  paragraph.  The 
term  mechanical  energy  includes  kinetic  energy  in  addition 
to  these. 

A  convenient  term  in  common  use,  closely  related  to 
energy,  is  the  term  work,  which  is  used  to  designate  the 
quantities  of  all  forms  of  energy  except  heat,  involved  in 
any  process  of  energy  transformation  or  transference ;  it 
therefore  signifies  the  quantities  of  the  various  forms  of 
mechanical  energy,  of  electrical  and  magnetic  energies,  and 
of  chemical  energy,  that  undergo  change  in  form  or  location 
in  any  process.  Thus,  it  is  customary  to  speak  of  the 
transformation  of  heat  into  work,  for  example,  by  the  steam 
engine  when  it  produces  motion  or  electrical  energy,  raises 
a  weight,  compresses  a  gas,  etc.,  and  in  general  of  the  pro- 
duction of  both  heat  and  work  in  the  surroundings  when 


72  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

a  body  parts  with  a  portion  of  its  energy,  the  two  terms 
heat  and  work  being  considered  to  embrace  all  forms  of 
energy  produced.  The  work  is  said  to  be  done  by  the 
body  which  parts  with  its  energy  upon  the  body  whose 
•energy  is  increased.  Thus,  work  is  done  by  an  explosive  at 
the  expense  of  its  chemical  energy  on  the  cannon  ball  which 
it  sets  in  motion,  and  by  the  steam  engine  at  the  expense  of 
the  heat  energy  of  the  steam  upon  the  body  which  it  raises. 
The  reason  for  thus  distinguishing  heat  from  other  forms  is 
the  existence  of  the  Second  Law  of  Energetics,  which  is 
discussed  below.  It  should  be  added  that  in  the  science  of 
Mechanics  the  term  work  is  used  in  a  more  specific  sense, 
namely,  to  designate  the  energy-change  corresponding  to  the 
displacement  of  a  force  through  a  distance ;  in  this  book, 
however,  it  will  be  used  only  in  its  general  sense. 

The  rate  at  which  energy  is  transferred  from  one  body 
to  another,  or  transformed  from  one  form  to  another,  is 
called  activity,  or  in  engineering  practice,  power,  when  the 
transference  or  transformation  is  of  such  a  character  that 
work  is  said  to  be  done.  That  is,  activity  or  power  is 
the  ratio  (dW  I  dr)  of  the  work  done  (dW)  to  the  time  (dr) 
in  which  it  is  done. 

Although  there  are  no  phenomena  that  furnish  direct 
manifestations  of  energy  unaccompanied  by  the  manifesta- 
tions of  matter  (§  7),  yet  there  are  phenomena  which  justify 
the  inference  that  energy  does  exist  unassociated  with 
matter.  Thus,  this  inference  is  to  be  drawn  from  the 
fact  that  portions  of  space  which  contain  no  matter,  under 
some  conditions  possess  the  power  of  producing  changes  in 
the  properties  of  bodies  that  are  introduced  into  the  space. 
In  illustration  of  this  statement,  the  following  phenomena 
may  be  cited :  any  non-transparent  body,  even  if  sur- 
rounded by  a  vacuum,  when  placed  in  a  sunbeam  acquires 
the  property  of  visibility ;  a  thermometer  with  blackened 
bulb  similarly  placed  indicates  by  the  extension  of  its 
mercury  column  a  rise  of  temperature ;  and  in  a  metal  ring 


GENERAL  PRINCIPLES  RELATING   TO  ENERGY.  73 

interrupted  by  a  small  spark-gap  electric  sparks  are  produced 
when  it  is  suitably  placed  near  a  circuit  having  a  spark-gap 
across  which  el-ectric  discharges  are  taking  place.  In  addition 
to  the  forms  of  energy  associated  with  matter,  which  have 
been  already  considered,  it  is  therefore  necessary  to  recog- 
nize another  form  in  which  energy  exists  unassociated  with 
matter.  This  form,  which  is  known  as  radiant  energy,  may 
be  defined  as  the  form  in  which  energy  is  transmitted  from 
one  body  to  another  through  space  without  the  mediation  of 
ordinary  matter.  Various  manifestations  of  it  may  be  dis- 
tinguished :  the  most  important  are  light,  radiant  heat,  and 
electromagnetic  radiations,  which  are  capable  of  producing 
in  bodies  the  properties  of  visibility,  increased  temperature, 
and  electrification  or  magnetization,  respectively.  They  are 
illustrated  by  the  three  examples  just  cited.  Besides  these, 
there  have  been  recently  discovered  other  more  special 
varieties  of  radiant  energy,  such,  for  example,  as  are  possessed 
by  the  so-called  Rontgen  Rays  and  Becquerel  Rays. 

*  The  general  definition  of  energy,  given  above,  as  that 
which  gives  rise  to  the  changes  in  properties  of  bodies,  is 
not  that  which  is  customary  in  treatises  on  physics,  where  it 
is  usually  defined  as  the  power  of  doing  work.  The  latter 
definition  is  open,  however,  to  the  objection  that  it  presup- 
poses a  definition  of  work,  which  cannot,  of  course,  in  this 
case  logically  be  defined  in  terms  of  energy,  as  has  been 
done  above,  but  must,  therefore,  be  defined  in  terms  of  some 
less  fundamental  concept,  that  of  force  being  usually  em- 
ployed. Another  definition,  which  has  recently  been  pro- 
posed, is,  energy  is  the  power  to  change  the  state  of  motion 
of  a  body.  While  the  same  objection  does  not  apply  to 
this,  both  these  special  definitions  have  the  disadvantage 
of  giving  undue  prominence  to  one  particular  form  of  en- 
ergy, and  tend  towards  an  exclusively  mechanical  inter- 
pretation of  natural  phenomena.  Another,  more  serious 
objection  to  these  definitions  is  the  fact  that  the  essence 
of  one  of  the  fundamental  laws  relating  to  energy  —  the 


74  GENERAL  PRINCIPLES  OF  PHYSICAL   SCIENCE. 

so-called  Second  Law  of  Energetics  —  is  that  heat  energy  and 
the  power  of  doing  work  actually  are  not  equivalent,  but  are 
sharply  differentiated.  The  usual  definition  is  therefore  a 
source  of  confusion.  The  general  definition  given  at  the 
beginning  of  this  section,  though  it  furnishes  a  sufficient 
criterion  for  detecting  the  presence  of  energy  in  any  body 
or  space,  is,  to  be  sure,  not  definite  enough  to  form  a 
basis  for  the  quantitative  measurement  of  energy;  further 
specifications  in  regard  to  the  latter  are  therefore  necessary. 
26.  The  Quantitative  Measurement  of  Energy.  —  In 
order  to  measure  quantities  of  energy  of  different  forms,  it 
is  necessary  to  convert  them  into  the  same  form,  and  com- 
pare the  quantities  of  it  so  produced  with  some  standard 
quantity  of  it  adopted  arbitrarily  as  a  unit.  Since  gravita- 
tion and  kinetic  energies,  unlike  the  other  forms,  can  be 
fully  expressed  in  terms  of  matter  and  space,  or  matter, 
space,  and  time,  without  the  introduction  of  any  new  con- 
cept, it  is  simplest  in  principle  to  adopt  a  unit-quantity  of 
one  of  these  forms  for  the  expression  of  quantities  of 
energy  in  general;  moreover,  the  pre-eminent  technical 
importance  of  mechanical  energy  furnishes  additional  reason 
for  the  adoption  of  some  form  of  it  as  a  standard  of  refer- 
ence. Some  unit  of  energy  based  upon  mechanical  relations 
has  therefore  been  universally  adopted  as  the  fundamental  one. 

The  centimeter-gram-second  unit  of  energy  is  called  the 
erg.  It  is  most  simply  defined  as  that  amount  of  energy 
which  is  equal  to  twice  the  kinetic  energy  possessed  by  a 
mass  of  one  gram  when  moving  with  a  velocity  of  one  centi- 
meter per  second.  As  this  unit  is  a  very  small  quantity,  a 
unit  ten  million  times  greater,  called  the  joule,  is  often  em- 
ployed ;  that  is,  i  joule  =  107  ergs.  The  reason  for  the 
apparently  irrational  definition  of  the  erg  as  twice  the  stated 
quantity  of  kinetic  energy  is  a  historical  one,  and  will  be 
explained  in  §  29. 

Units  of  heat  energy  are,  however,  also  extensively  em- 
ployed, especially  by  reason  of  the  fact  that  other  forms  of 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.  75 

energy  are  readily  and  completely  transformed  into  heat, 
so  that  it  is  the  form  which  is  in  general  best  adapted  for 
the  experimental  determination  of  quantities  of  energy. 
The  unit  of  heat  will  be  here  defined  as  the  quantity  of 
heat  required  to  raise  the  temperature  of  I  gram  of  water 
from  17°  to  1 8°.  This  unit  is  called  the  calorie  at  IJ.f 
(cal.),  and  it  is  the  one  that  will  be  always  used  in  this  book. 
It  is  to  be  noted,  however,  that  usage  unfortunately  varies 
with  respect  to  the  temperature  adopted  in  the  definition, 
and  that,  therefore,  the  calorie  used  by  different  scientists 
varies  somewhat  in  value  (decreasing  by  0.03  per  cent 
per  degree  between  15°  and  25°).  Moreover,  certain  other 
heat  units  are  in  common  use.  Of  these  will  be  here 
mentioned  only  the  mean  calorie,  which  is  one  one-hundredth 
part  of  the  heat  required  to  raise  the  temperature  of  i  gram 
•of  water  from  o°  to  100°.  According  to  recent  determi- 
nations, the  mean  calorie  differs  from  the  calorie  at  17.5° 
by  less  than  o.  i  per  cent. 

The  relation  between  the  mechanical  and  thermal  units 
is  evidently  of  very  great  importance,  and  it  has  been  de- 
termined with  great  care  by  several  experimenters.  The 
number  of  units  of  mechanical  energy  which  correspond  to 
one  unit  of  heat  energy  is  called  the  mechanical  equivalent 
of  heat  (J).  Its  value  in  the  centimeter-gram-second  system 
has  been  found  experimentally  to  be  41  840000.  That  is, 
i  calorie  =4. 1 84  X  io7  ergs  =  4. 1 84  joules,  or  approximately 
4.2  joules. 

It  should  be  added  that  it  is  now  becoming  a  common 
practice  to  express  quantities  of  heat  in  joules  instead  of 
calories,  the  direct  results  of  calorimetric  measurements  being 
reduced  to  the  mechanical  unit  with  the  help  of  the  mechan- 
ical equivalent  of  heat. 

The  unit  of  activity  or  power  in  the  centimeter-gram- 
second  system  is  that  involved  when  one  erg  of  work  is  done 
per  second.  No  name  has  been  given  to  this ;  for  a  unit  ten 
million  times  as  large  is  far  more  commonly  employed.  This 


7C  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

latter  unit  is  called  the  watt,  and  is  the  rate  at  which  work 
is  done  when  one  joule  of  energy  is  transferred  or  trans- 
formed per  second. 

27.  The  Law  of  the  Conservation  of  Energy,  or  the 
First  Law  of  Energetics.  —  The  essential  idea  involved  in 
the  concept  of  energy  and  throughout  the  above  considera- 
tions relating  to  it  is  the  constancy  of  a  quantity  which 
is  involved  in  all  the  changes  taking  place  in  the  universe ; 
and  this  is  often  explicitly  expressed  by  the  statement 
that  energy  is  neither  created  nor  destroyed  in  any  process 
whatever.  This  statement  is  called  the  Law  of  the  Conser- 
vation of  Energy  or  the  First  Law  of  Energetics,  the  name 
Energetics  being  applied  to  that  branch  of  science  which 
treats  of  the  general  principles  relating  to  energy.  This  law 
may  also  be  stated  more  concretely  as  follows  :  When  a 
quantity  of  energy  disappears  at  any  place,  a  precisely 
equal  quantity  of  energy  simultaneously  appears  at  some  other 
place  or  places,  and  when  a  quantity  of  energy  disappears  in 
any  form,  a  precisely  equal  quantity  of  energy  simultaneously 
appears  in  some  other  form  or  forms ;  equal  quantities  of 
energy  of  different  forms  being  understood  to  be  such  quan- 
tities as  produce  the  same  effect  (for  example,  in  modifying 
motion  or  raising  temperature)  when  converted  into  the 
same  form. 

This  law  is  to  be  regarded  as  one  of  the  most  certain,  as 
well  as  one  of  the  most  fundamental,  laws  of  science.  The 
exactness  of  it  has  been  established  by  a  considerable  number 
of  careful  quantitative  investigations  made  for  the  purpose. 
Thus,  the  various  determinations  of  the  mechanical  equivalent 
of  heat  have  proved  that  a  definite  quantity  of  mechanical 
energy,  such  as  that  possessed  by  a  raised  weight,  when 
transformed  into  heat,  always  gives  rise  to  a  definite  quantity 
of  heat,  whatever  may  be  the  process  used  for  the  transfor- 
mation ;  for  example,  in  one  method  a  weight  while  gradually 
falling  kept  a  paddle-wheel  in  motion  which  by  its  friction 
against  the  water  of  a  calorimeter  produced  the  heat,  while 


GENERAL  PRINCIPLES  RELATING   TO  ENERGY.          77 

in  another  method  a  weight  was  allowed  to  fall  freely  and  to 
strike  upon  a  mass  of  lead,  which  became  heated  as  a  result 
of  its  compression.      Such  determinations  have  also  shown 
that   those   quantities   of  electrical   energy,  kinetic  energy, 
gravitation  energy,  and  volume  energy  (of  compressed  gases) 
which  are  transformable  into  one  another  produce  equal  quan- 
tities of  heat,  thus  proving  that  there  is  no  change  in  the  total 
energy  when  these  four  forms  are  converted  into  one  another. 
The  law  is   also   confirmed   by   the   correspondence   of  the 
conclusions  drawn  from  it   with   well-established   facts   and 
principles.      Among   these   may   be  mentioned  as  most  im- 
portant the  following  principle,  which  is  a  conclusion  based 
upon  the  failure  of  innumerable  attempts  to  produce  a  con- 
trary result :     The  production  of  an   unlimited  amount  of 
work  by  a  machine  or  arrangement  of  matter  which  receives 
no  energy  from  the  surroundings  is  an  impossibility.     An 
ideal  process  like  that  here  stated  to  be  impossible  of  realiza- 
tion is  sometimes  called  perpettial  motion  of  the  first  kind. 
It  is  often  desirable  in  applications  of  the  Laws  of  Ener- 
getics to  differentiate  the  energy-changes  occurring  within  a 
definite  body  or  group  of  bodies,  that  is,  within  the  system 
under   consideration,  from   the  related   changes  which  may 
simultaneously  occur  in  the  surroundings,  and  to  differenti- 
ate  further   the    energy-changes   in   the   surroundings   into 
work  and   heat-changes,  as   described   in    §  25.     Thus,  the 
total  energy  within   a   definite  system  (not  including,  how- 
ever,  its    kinetic  energy)  is   commonly  called    its   internal 
energy  ([/);  and  the  requirement  of  the  First  Law  that  the 
difference  (U*  —  6^)  in  its  values  after  and  before  any  change 
be  equal  to  the  quantity  of  heat  (Q)  absorbed  from  the  sur- 
roundings diminished  by  the  work  ( W)  done  by  the  system 
on  the  surroundings  (called  the  external  work),  is  expressed 
by  the  equation : 

K  —  tfi  =  Q  —  W. 

It  is  to  be  noted  that  here  and  always  in  this  book  Q  repre- 


78  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

sents  the  heat  absorbed  by  the  system,  and  W  the  work  pro- 
duced in  the  surroundings,  and  therefore  that  the  numerical 
value  of  Q  is  positive  when  heat  is  actually  absorbed  and 
negative  when  it  is  evolved,  and  that  that  of  W  is  positive 
when  work  is  done  by  the  system  upon  the  surroundings  and 
negative  when  done  by  the  surroundings  upon  the  system. 

The  value  of  the  increase  in  the  internal  energy  is 
evidently  equal  to  the  heat  absorbed  by  the  system  when 
the  same  change  in  state  takes  place  under  such  conditions 
that  no  external  work  is  done ;  that  is,  Uz  —  U\  =  Q,  when 
W-=Q\  and  it  is  commonly  so  determined. 

Consider  as  an  example  the  change  consisting  in  the 
volatilization  at  100°  of  I  gram  of  liquid  water  (whose  vol- 
ume is  1.043  ccm.)  to  form  vapor  (volume,  1661  ccm.)  against 
the  constant  pressure  of  one  atmosphere.  Since  the  heat 
absorbed  by  the  system  during  the  change  (the  ordinary 
heat  of  vaporization)  is  537  cal.,  and  the  work  done  as  the 
result  of  the  expansion  can  be  calculated  (see  §  30)  to  be  168 
joules,  the  values  in  the  above  equation  become : 

£/2  —  #i  —  (537  x  4.184)  —  168  =  2082  joules. 

This  last  value,  the  increase  in  the  energy  of  the  water  as  a 
result  of  its  vaporization,  is  equal  to  the  heat  that  would  be 
absorbed  if  the  water  were  vaporized  within  a  vessel  of  con- 
stant volume  ;  for  then  no  external  work  would  be  done. 

Another  example  of  quite  a  different  kind  is  that  of  a 
chemical  change  which  is  accompanied  by  the  production  of 
electrical  energy.  Consider  a  system  consisting  of  a  silver 
plate  immersed  in  a  silver  nitrate  solution  and  a  copper  plate 
in  a  copper  nitrate  solution,  the  two  solutions  being  in  contact 
with  each  other,  and  the  two  plates  being  connected  with 
wires  by  which  the  electrical  energy  produced  is  transferred 
to  the  surroundings ;  consider  further  that  the  chemical 
change  takes  place  which  is  expressed  quantitatively  as  well 
as  qualitatively  by  the  equation  : 

Cu  +  2AgN03  =  Cu(N03)2  +  2Ag. 


GENERAL  PRINCIPLES  RELATING   TO  ENERGY.          79 

Since  this  change  is  known  to  evolve  8920  cal.  when  taking 
place  under  the  stated  conditions  (that  is,  so  that  electrical 
energy  is  produced),  and  to  evolve  30040  cal.  when  no  ex- 
ternal work  is  done  (for  example,  when  the  plate  of  copper  is 
placed  directly  in  the  silver  nitrate  solution),  the  equation, 
Uz  —  tfi  =  Q  —  W,  becomes,  (—  30040)  =  (—  8920)  —  EE , 
or  EE  =  2 1 1 20  cal.,  where  EE  represents  the  electrical  energy 
produced. 

28.  The  Factors  of  Energy  in  General. —  Experience 
shows  that  whether  or  not  a  transfer  of  any  form  of 
energy  from  one  place  to  another  takes  place,  and  in  which 
direction  it  takes  place,  is  not  determined  by  differences 
in  the  quantities  of  the  energy  in  the  two  places.  It  is 
therefore  necessary  to  ascribe  to  every  form  of  energy  a 
characteristic  property  which  gives  rise  to  this  phenome- 
non of  transference.  This  property  is  called  the  intensity 
of  the  energy,  and  is  assumed  to  have  a  greater  value  at  the 
place  where  the  quantity  of  energy  diminishes,  and  a  smaller 
one  at  the  place  where  it  increases.  Therefore,  wherever  a 
transference  of  energy  takes  place,  a  difference  of  intensity 
must  exist. 

For  example,  when  a  small  vessel  of  water  is  placed 
within  a  larger  one,  it  often  happens  that  no  transfer  of  heat 
takes  place,  although  the  amount  of  heat  contained  in  the 
larger  vessel  may  be  much  greater ;  while  in  other  cases  it 
will  be  found  that  heat  passes  from  the  smaller  to  the  larger 
vessel,  or  the  reverse,  making  it  evident  that  it  is  not  the 
quantity  of  heat,  but  some  property  of  it,  that  determines 
its  transference ;  this  property  is  in  this  case  the  familiar 
one  called  temperature.  Similarly,  if  two  reservoirs  of  gas 
are  connected  with  the  opposite  ends  of  a  closed  cylinder 
containing  a  movable  piston,  it  is  found  that  whether  or  not 
motion  of  the  piston  takes  place,  or  in  which  direction  it 
takes  place,  does  not  depend  on  the  quantities  or  the 
volumes  of  the  two  gases,  or  on  the  quantities  of  energy 
contained  in  them,  but  on  the  particular  property  which  is 


80  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

called  pressure.  Temperature  and  pressure  are,  then,  the 
intensities  of  heat  and  volume  energies,  respectively. 

It  is  always  found  that,  other  things  being  equal,  the 
quantity  of  energy  involved  in  any  transference  of  energy 
is  greater,  the  greater  the  difference  in  intensity.  There- 
fore, if  an  arbitrary  scale,  such  as  has  been  already  described 
in  the  case  of  temperature,  be  adopted  for  the  measurement 
of  intensity-differences,  a  quantity  of  energy  of  whatsoever 
form  may  be  regarded  as  the  product  of  two  numerical 
factors  :  one  of  these  is  the  value  of  the  intensity-difference, 
and  the  other  is  the  quantity  by  which  this  must  be  multi- 
plied in  order  that  the  product  may  equal  the  quantity  of 
energy.  This  latter  factor  is  called  the  capacity-factor  of 
the  energy.  Its  numerical  value  will  evidently  be  equal 
to  the  quantity  of  energy  transferred  when  the  intensity- 
difference  is  constant  and  equal  to  unity. 

Although  this  method  of  determining  the  factors  of 
energy  is,  abstractly  considered,  the  rational  one,  the  reverse 
method  is  often  followed,  for  the  reason  that  the  capacity- 
factor  of  some  forms  of  energy  is  more  familiar  and  more 
directly  measurable  than  the  intensity-factor.  The  method 
actually  employed  is  therefore  the  following  one.  It  is 
almost  always  possible  to  detect  at  once  some  one  definite 
well-known  property  on  which  the  magnitude  of  the  energy 
in  question  depends ;  for  example,  the  energy  of  a  moving 
body  is  obviously  dependent  on  its  rate  of  motion  or  ve- 
locity, that  of  a  suspended  body  on  its  distance  above  the 
earth's  surface,  that  of  a  hot  body  on  its  temperature.  A 
method  of  quantitative  measurement  of  this  property  is 
agreed  upon,  and  the  property  is  adopted  as  one  of  the 
factors  of  the  energy.  The  other  factor  is  thereby  deter- 
mined; for  it  is,  by  definition,  that  quantity  which  when 
multiplied  with  the  value  of  the  factor  already  adopted  will 
produce  the  energy.  It  is  generally  found  that  one  of  the 
two  factors  is  the  intensity  of  the  energy,  or  that  property 
of  it  which  determines  its  transference.  If  such  did  not 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.  81 

prove  to  be  the  case,  another  method  of  resolution  into 
factors  would  be  attempted.  Examples  of  two  possible 
methods  of  resolution  into  factors  are  given  below  in  the 
cases  of  kinetic  and  gravitation  energies. 

These  considerations  are  made  clearer  and  more  defi- 
nite by  the  following  mathematical  expression  of  them.  In 
accordance  with  the  preceding  statements,  the  relation 
between  the  quantity  (E)  of  any  form  of  energy  and  the 
values  ( /  and  c )  of  its  intensity  and  capacity  factors  is 
expressed  by  the  equation  : 

E^ic. 

That  is,  the  value  of  either  factor  is  the  ratio  of  the  value 
of  the  energy  to  the  corresponding  value  of  its  other  factor. 

The  differential  form  of  this  equation, 

dE  =  d(ic)  =  idc  +  c  di, 

is,  however,  of  more  general  applicability ;  for  in  most  cases 
it  is  not  possible  to  measure  the  absolute  quantity  of  a  form 
of  energy  present  in  a  system,  although  changes  in  its 
quantity  may  be  readily  accessible  to  measurement  It 
follows  from  this  equation  that : 

dE  =  idc,  when  i  is  constant ;  and 
dE  =  c  dit  when  c  is  constant. 

These  two  equations  are  applied  in  three  different  ways  in 
determining  the  relation  between  a  form  of  energy  and  its 
factors.  When  the  energy-change  of  the  form  in  question 
is  capable  of  direct  measurement,  some  principle  is  adopted 
either  for  the  measurement  of  the  absolute  value  of  one  of 
the  factors  (i  or  c),  whereby  the  change  in  the  value  of  the 
other  factor  (dc  or  di)  becomes  determined,  or  for  that  of 
the  value  of  the  change  in  one  factor  (dc  or  di)t  whereby 
the  absolute  value  of  the  other  factor  becomes  determined ; 
or  thirdly,  when  the  energy-change  is  not  directly  measur- 
able (owing  perhaps  to  its  being  necessarily  accompanied 
by  a  change  in  some  other  form  of  energy  than  that  in 
question),  principles  are  adopted  for  the  measurement  both 


82  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

of  the  absolute  value  of  one  factor  and  of  the  change  in 
value  of  the  other  factor. 

Numerous  applications  of  these  principles  are  presented 
in  the  following  sections,  in  which  the  factors  of  the  various 
forms  of  energy  are  considered. 

29.  The  Factors  of  Kinetic  and  Gravitation  Ener- 
gies. Definitions  of  Force.  —  The  kinetic  energy  of  a 
body  is  obviously  dependent  on  its  velocity  (u).  Measure- 
ments of  the  quantities  of  energy  possessed  by  the  same 
body  when  moving  at  different  velocities  (made,  for  ex- 
ample, by  determining  the  rises  of  temperature  produced 
in  a  definite  quantity  of  water  when  the  motion  is  destroyed 
by  friction)  show,  however,  that  these  quantities  are  not 
proportional  directly  to  the  velocity,  but  to  its  square ;  the 
velocity-square  is  therefore  one  of  the  factors  of  kinetic 
energy.  Kinetic  energy  does  not  depend  upon  this  alone, 
however  ;  for  different  bodies  moving  with  the  same  velocity 
are  found  to  have  different  amounts  of  energy ;  this  other 
factor  upon  which  the  magnitude  of  the  energy  depends,  is 
called  mass  (m),  as  already  stated  in  §  7.  Finally,  it  is  to 
be  noted  that  the  definition  adopted  for  the  unit  of  energy, 
the  erg,  causes  the  introduction  of  the  coefficient  J  into 
the  relation  between  kinetic  energy  (EK)  and  its  factors. 
These  principles  and  definitions  are  expressed  in  the 
equation : 

It  is  to  be  noted  that  velocity  is  a  quantity  which 
requires  for  its  complete  definition  not  only  the  specification 
of  a  magnitude,  but  also  that  of  a  direction,  which  last  is 
usually  expressed  with  reference  to  arbitrary  co-ordinate 
axes,  and  in  such  a  manner  that  velocities  in  opposite  direc- 
tions are  represented  by  opposite  signs.  In  the  case  of 
velocity-square,  however,  and  therefore  in  that  of  kinetic 
energy,  the  factor  expressing  the  direction  is  eliminated 
(since  the  sign  of  a  square  is  positive,  whatever  be  the  sign 
of  the  quantity).  Now,  since  bodies  change  their  relative 


GENERAL   PRINCIPLES  RELATING    TO  ENERGY.          83 

positions  when  they  possess  velocities  differing  either  in 
magnitude  or  direction,  it  is  evident  that  velocity,  and  not 
velocity-square,  is  to  be  regarded  as  the  intensity-factor  of 
kinetic  energy.  Its  capacity-factor  then  becomes  the  prod- 
uct of  the  mass  into  the  velocity  (mu),  which  product  is 
commonly  called  momentum.  This  method  of  resolution  is 
expressed  by  the  equation,  EK  =  \(m  u)  u.  These  consider- 
ations illustrate  the  fact  that  the  more  obvious  method  of 
resolving  an  energy-quantity  (that  first  presented  in  this 
case)  may  not  yield  a  factor  which  has  the  characteristics  of 
an  intensity-factor.  They  also  illustrate  the  application  of 
the  general  equation,  E  =  ic,  to  the  determination  of  the 
relation  of  a  form  of  energy  to  its  factors,  it  being  known  in 
this  case  that  i  is  equal  to  u,  for  the  measurement  of  which 
units  were  described  in  §  5,  and  that  E  =  ^  m  u*,  whereby 
c  becomes  determined. 

The  most  obvious  and  simple  factor  on  which  the  quan- 
tity of  gravitation  energy  depends,  is  the  distance  between 
the  bodies  possessing  it.  Newton's  Law  of  Gravitation, 
when  expressed  in  terms  of  energy  instead  of  force,  shows, 
however,  that  the  gravitation  energy  (E^  possessed  by  two 
bodies  is  not  directly  proportional  to  the  distance  (/)  be- 
tween them,  but  that  it  varies  with  the  distance  in  the  way 
expressed  by  the  following  equations : 


(±-     -i- 

U     A 


where  m\  and  m'z  are  constants  depending  solely  on  the  two 
bodies  ;  and  where  EG^  and  EG^  are  the  quantities  of  gravita- 
tion energy  possessed  by  them  when  at  the  distances  /x  and  /2, 
respectively.  It  is  evident  now  that,  if  the  function  of  the 
distance  (-— )  were  adopted  as  one  factor  of  the  energy, 
the  product  of  the  quantities  m\  and  m\  would  be  the  other 
factor.  These  quantities  are  a  permanent  property  of  the 


84  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

individual  bodies,  entirely  independent  of  their  variable 
properties,  and  also  of  the  distance  between  them  ;  it  is  this 
property  which  was  called  in  §  7  the  capacity  of  bodies  for 
gravitation  energy.  By  Newton's  Law  of  Gravitation  these 
quantities  m\  and  m!*  are,  as  stated  in  §  7,  strictly  pro- 
portional to  the  masses  m±  and  m^  of  the  bodies  ;  that  is, 
m\  m\  =j  mi  m2,  where  j  is  the  so-called  gravitation- 
constant. 

This  method  of  resolution  of  gravitation  energy  into 
factors  is  of  importance  in  its  relation  to  the  concept  of 
matter ;  for  it  shows  very  clearly  that  the  property  expressed 
by  the  quantities  m\  and  m'*  derived  from  a  consideration  of 
the  gravitation  energy  of  bodies  is  just  as  fundamental  a 
one  as  that  of  mass,  which  is  derived  by  an  entirely  similar 
consideration  of  their  kinetic  energy.  From  the  energy 
point  of  view  this  method  of  resolution  is  not  satisfactory, 
however ;  for  neither  of  the  factors  has  the  character  of 
an  intensity,  since  neither  the  distance  between  the  bodies, 
nor  any  inherent  property  of  the  bodies  themselves,  alone 
determines  the  degree  of  their  tendency  to  approach  one 
another,  which  is  found  to  be  dependent  both  on  the  bodies 
themselves  and  the  distance  between  them.  It  is  found, 
however,  that,  if  the  distance  itself,  instead  of  any  function 
of  it,  is  adopted  as  one  of  the  factors  of  the  energy,  then 
the  quantity  by  which  this  must  be  multiplied  to  produce 
the  energy  does  have  the  characteristics  of  an  intensity- 
factor.  This  quantity  is  called  force, — a  term,  however, 
which  is  not  only  used  to  designate  the  intensity-factor  of 
the  gravitation  energy  possessed  by  two  bodies,  but  is  also 
employed,  in  a  manner  to  be  now  described,  in  a  much  more 
general  sense  and  with  reference  to  other  forms  of  distance 
energy. 

Whenever  in  displacing  a  body  through  a  distance  (dl) 
in  any  direction,  an  increase  in  its  distance  energy  (dE'fr 
takes  place  (whereby,  of  course,  an  equivalent  quantity  of 
work  must  be  done  upon  it),  a  force  (F)  is  said  to  be  acting 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.  86 

upon  the  body  in  the  opposite  direction,  and  its  magnitude 
is  defined  to  be  equal  to  the  ratio  of  the  change  in  distance 
energy  to  the  change  in  distance;  that  is,  F  =  dEi/  dl. 
A  negative  value  of  dEh  that  is,  an  energy-decrease, 
signifies  a  force  acting  in  the  direction  of  the  body's  dis- 
placement. 

Employing  this  concept  of  force,  changes  in  distance 
energy  are  expressed  in  a  most  general  manner  by  the 
equations  : 

dEl  =  Fdl,  and  Elt  —  E/t  = 

The  last  equation  becomes,  E^  —  Elv  =  F  (/a  —  /J,  when 
the  force  is  constant  throughout  the  distance  traversed. 

This  method  of  resolving  distance  energy  into  factors 
and  of  defining  force  is  evidently  an  application  of  the  gen- 
eral equation,  i  =  dE  /  dc,  to  a  case  where  the  quantities  dE 
and  dc  have  been  first  independently  defined. 

It  follows  from  this  definition  of  force  and  the  induc- 
tively established  laws  of  gravitation,  which,  as  above  stated, 
are  expressed  by  the  equations  : 


T  r" 

dE 


If       Jl  1      \  ^^   1  Wt  a      JT  *  Wll  Wt*     jr 

=  m\  m\  <t(-  j\  =  —  i^-!  dl  =j  -*_»  dl, 


that  for  the  force  of  gravitation,  FG>  the  following  relation 
holds  true : 


where  j  is  the  gravitation-constant.  This  last  equation  is 
evidently  the  mathematical  expression  of  the  laws  of  gravi- 
tation in  the  form  in  which  they  were  stated  by  Newton. 
Since  distance  energy  is  completely  transformed  into 
kinetic  energy  with  great  readiness,  as  in  the  case  of  a  body 
falling  towards  the  earth,  a  secondary  definition  of  force  can 
be  based  on  the  relation  between  these  two  forms  of  energy. 
This  is  derived  immediately  from  the  Law  of  the  Conser- 
vation of  Energy,  which  requires  that  a  decrease  in  the 


«6          GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

quantity  of  the  one  form  of  energy  be  accompanied  by  an 
equal  increase  in  the  quantity  of  the  other  form,  provided, 
of  course,  no  third  form  of  energy  is  produced ;  that  is  : 

Fdl  —  d  (^  m  #2)  =  m  u  du, 

where  F  represents  a  force  which,  in  acting  upon  a  body  of 
mass  m  moving  with  a  velocity  u,  in  the  direction  of  its 
motion  for  the  time  dry  during  which  it  traverses  the  distance 
dl,  produces  an  increase  in  its  velocity  du.  From  this  equa- 
tion it  follows,  since  u  =  dl  I  dr  and  a  =  du  /  dr  (§  5),  that 

du 

F  =  m  —  =  m  a  ; 
dr 

that  is,  force  is  measured  by  the  product  of  the  mass  of  a 
t>ody  into  the  acceleration  which  the  force  produces  in 
its  motion. 

This  last  definition  is  the  one  originally  adopted  for 
force  before  the  development  of  the  doctrine  of  energy, 
and  it  is  still  retained  as  a  primary  one  in  most  works  on 
physics,  energy  being  subsequently  defined  as  the  power 
of  doing  work,  which  in  turn  is  defined  in  terms  of  force 
and  distance.  This  procedure  is  liable  to  lead  to  a  confu- 
sion of  the  fundamental  with  the  derived  concept;  and  it  is 
therefore  necessary  to  emphasize  the  relation  and  distinction 
between  energy  and  force.  Energy  is  the  ultimate  cause 
which  produces  the  tendency  in  bodies  to  approach  or  re- 
cede from  one  another,  just  as  it  produces  the  tendency  to 
all  other  changes ;  force  is  the  characteristic  which  energy 
manifests  when  it  tends  to  cause  bodies  to  approach  or  recede 
from  one  another.  Energy  is  an  indestructible  quantity; 
force  is  a  changeable  quality  of  energy,  sometimes  exhibited 
by  it  and  sometimes  not. 

In  accordance  with  the  last  equation,  the  value  of  the 
force  FG  acting  between  the  earth  and  a  body  of  mass  m, 
which  force  is  called  the  force  of  gravity,  is  given  by  the 
equation,  FG  =  mg,  in  which  g  represents  the  acceleration 
of  its  motion  which  the  body  experiences  when  falling 


GENERAL  PRINCIPLES  RELATING   TO  ENERGY.  87 

towards  the  earth  uninfluenced  by  any  other  cause  than  the 
force  of  gravity  itself.  The  value  of  g  is  independent  of 
the  mass  of  the  falling  body,  but  varies  somewhat  with  its 
position. 

The  fundamental  unit  employed  for  the  measurement 
of  force  is  called  the  dyne ;  it  is  the  force  which  is  acting 
continuously  upon  a  body  when  an  increase  or  decrease  of 
one  erg  in  its  distance  energy  takes  place  for  each  centi- 
meter of  distance  traversed ;  or,  in  accordance  with  the 
secondary  definition  of  force,  it  is  the  force  which  when 
exerted  on  a  mass  of  one  gram  for  one  second  causes  its 
velocity  to  be  increased  by  the  amount  of  one  centimeter 
per  second.  Since  the  increase  of  velocity  that  takes  place 
each  second  in  the  downward  motion  of  a  freely  falling  body 
at  any  place  is  g  centimeters  per  second,  the  dyne  is  evi- 
dently one  ^-th  part  of  the  force  with  which,  at  that  place, 
a  mass  of  one  gram  tends  to  approach  the  earth.  This  last 
force  is  known  as  a  force  of  one  grant  in  the  so-called 
Gravitational  System  of  Units.  For  exact  scientific  pur- 
poses this  system  is  an  unsatisfactory  one,  since  the  value 
of  the  unit  of  force  is  different  at  different  places  upon  the 
earth ;  it  will  therefore  not  be  employed  in  this  book.  It  is 
important  to  note,  however,  that  the  value  of  a  force 
expressed  in  grams  at  any  place  is  converted  into  the  value 
of  it  in  dynes  by  multiplying  by  the  value  of  g  at  that  place  ; 
also,  that  the  value  of  g  at  the  sea-level  in  a  latitude  of 
45°  is  980.7,  or  approximately  980,  using  as  units  the  cen- 
timeter and  second,  and  that  variations  of  altitude  not  ex- 
ceeding 3000  meters  and  of  latitude  not  exceeding  10° 
affect  this  value  by  less  than  o.i  per  cent. 

The  definition  of  the  unit  of  energy,  the  erg,  was 
originally  based  on  that  of  the  unit  of  force,  the  dyne ;  the 
latter  having  been  first  defined  in  accordance  with  the 
equation,  F=  m  a,  and  the  former  then  becoming  determined 
by  the  equation,  dE  =  Fdl.  This  method  of  definition 
makes  the  kinetic  energy  expressed  in  ergs  equal  to  one-half 


88  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

the  value  of  the  product  m  u*,  as  is  shown  by  the  series  of 
relations : 

dEK  =  Fdl  =madl—m dl  =  m  u  du, 

dr 

whence  it  follows,  by  integrating  under  the  assumption  that 
EK  =  o  when  u  =  o,  that  EK  =  J  m  u*.  If  then  the  unit 
of  energy  is  to  be  denned  in  terms  of  kinetic  energy,  and  is 
to  have  the  same  magnitude  as  when  defined  in  terms  of 
force  in  the  manner  just  described,  it  must,  as  was  done  in 
§  26,  be  defined  as  twice  the  energy  possessed  by  a  mass  of 
one  gram  when  moving  with  a  velocity  of  one  centimeter 
per  second ;  for  it  follows  from  the  last  equation  that 
EK  =  \  when  m  =  I  and  u  =  I. 

30.  The  Factors  of  Surface,  Volume,  and  Elastic 
Energies. —  Since  changes  in  cohesion  and  disgregation  ener- 
gies, the  forms  of  energy  which  are  associated  with  the 
particles  of  bodies,  can  not  as  a  rule  be  separately  measured, 
attention  will  be  here  confined  to  the  relation  between  the 
external  changes  in  bodies,  that  is,  between  the  changes  in 
their  surface,  volume,  and  form,  and  the  energy-changes  that 
may  be  thereby  produced  in  the  surroundings.  In  accordance 
herewith,  it  is  convenient,  as  stated  in  §  25,  to  employ  the 
terms  surface,  volume,  and  elastic  energies,  to  designate  the 
powers  of  doing  external  work  which  bodies  possess  in  virtue 
of  their  tendency  to  undergo  changes  in  surface,  volume,  and 
form,  respectively.  These  energies  are  therefore  measured 
by  placing  the  changes  in  them  accompanying  a  definite 
change  of  surface,  volume,  or  form,  equal  to  the  maximum 
quantity  of  external  work  which  the  change  can  produce. 
This  quantity  of  work  is,  moreover,  usually  determined  by  first 
measuring  the  force  that  must  be  externally  applied  to  the 
body,  in  order  to  compensate  the  internal  force  outwardly 
manifested  by  it  and  thus  prevent  the  change  from  taking 
place,  and  then  multiplying  this  external  force  by  the  distance 
through  which  it  is  displaced. 

It  is  evident,  when  a  definite  change  in  surface,  volume, 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.  89 

or  form  takes  place  under  this  condition  of  substantially  com- 
pensated external  and  internal  forces,  that  the  maximum 
amount  of  work  is  obtained  from  the  change :  for,  if  the  ex- 
ternal force  applied  were  not  less  in  value  by  at  least  an 
infinitesimal  amount  than  the  internal  force,  the  change  could 
not  take  place ;  if,  on  the  other  hand,  it  were  less  by  a 
finite  amount,  the  external  force  displaced,  and  there- 
fore the  work  done,  would  be  less  than  it  might  be. 
A  similar  consideration  will  show  that  this  maximum  quan- 
tity of  work  is  substantially  identical  with  the  minimum 
quantity  of  work  by  the  expenditure  of  which  the  reverse 
change  in  the  surface,  volume,  or  form  of  the  body  can  be 
brought  about. 

It  will  be  clear  from  these  statements  that  a  decrease 
in  the  surface,  volume,  or  elastic  energy  of  a  system  is  not 
necessarily  accompanied  by  the  production  of  an  equivalent 
quantity  of  work,  or  indeed  of  any  quantity  of  work,  in  the 
surroundings,  since  the  internal  force  may  be  entirely  uncom- 
pensated,  as  in  the  expansion  of  a  gas  into  a  vacuum,  or  the 
release  of  a  stretched  spring.  It  is  true,  moreover,  that  the 
decrease  in  the  quantities  of  these  energies,  except  in  the 
case  in  which  the  maximum  amount  of  external  work  is  done, 
is  in  general  not  accompanied  by  an  equivalent  increase  in  the 
quantity  of  any  form  of  energy,  the  deficit  of  work  not  being 
compensated  by  a  corresponding  production  of  heat :  thus,  it 
will  be  seen  later  that  in  the  expansion  of  a  perfect  gas  into  a 
vacuum,  there  is  not  only  no  production  of  work,  but  also 
none  of  heat ;  yet  the  volume  energy  of  the  gas,  or  its  power 
of  doing  external  work  in  virtue  of  its  tendency  to  undergo 
volume-changes,  has  obviously  decreased*  This  apparent 
contradiction  with  the  First  Law  of  Energetics  disappears 
when  it  is  recognized  that  quantities  of  surface,  volume,  or 
elastic  energies  do  not  have  the  same  significance  as  do 
quantities  of  one  of  the  forms  of  energy,  such  as  kinetic, 
gravitation,  or  cohesion  energy,  whose  disappearance  is 
necessarily  attended  by  the  appearance  of  an  equivalent 


90  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

quantity  of  some  other  form  of  energy ;  for  quantities  of  the 
first-named  energies  are  defined  to  be  those  portions  of  the 
total  internal  energy  of  bodies  which  represent  the  power  of 
producing  external  work  when  changes  in  the  surface,  volume, 
or  form  of  the  bodies  take  place ;  and  a  decrease  in  this 
specific  power  in  the  body  is  not  necessarily  accompanied  by 
a  corresponding  decrease  in  its  power  of  producing  those 
other  effects  which  are  a  measure  of  its  total  internal  energy. 
This  distinction  will  become  clearer  in  connection  with  the 
discussion  of  the  Second  Law  of  Energetics. 

As  has  been  already  stated,  the  surfaces  of  liquids  tend 
to  diminish  in  extent  and  are  capable  of  doing  work  in  virtue 
of  this  tendency;  they  possess,  therefore,  surface  energy. 
Changes  in  the  quantity  of  this  energy  are  readily  measured 
in  the  case  where  a  surface  changes  in  extent  in  one  direction 
only.  Consider,  for  example,  that  a  surface  having  the  length 
/.,,  in  one  direction,  and  the  length  ly  in  a  direction  at  right 
angles  to  the  first,  is  increased  in  its  extent  in  the  latter  direc- 
tion by  an  amount  dly ,  and  that  the  external  force  Ft  acting  in 
this  direction  must  be  applied  to  the  line  lx  at  right  angles  to 
it,  in  order  to  compensate  the  internal  surface-force  Fs  acting 
upon  that  line  in  the  opposite  direction,  and  to  cause  the  ex- 
tension of  surface  to  take  place.  The  surface-force  acting  upon 
the  unit  of  length,  which  force  is  called  the  surface-tension  7, 
is  then  Fs  /  lx;  and  the  change  of  surface  ds  is  lx  dly.  The 
work  ( — dW)  done  upon  the  surface,  or  the  increase  in  the 
surface  energy  dEs,  is  then  given  by  the  equations : 

-dW=^dEs=Fl  dly  =  Fs  dly  =  i  lx  dly  =  7  ds; 

that  is,  it  is  equal  to  the  product  of  the  surface-tension  into 
the  increase  of  surface  produced. 

This  resolution  of  surface  energy  into  factors  is  an  appli- 
cation of  the  equation  dE  =  idc  to  a  case  where  i  and  dc  are 
first  determined,  the  energy-change  being  defined  to  be  equal 
to  their  product. 

A  concrete  illustration  involving  the  principle  of  one  of 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY. 


01 


the  common  methods  by  which  surface-tension  is  experimen- 
tally determined  will  serve  to  make  clearer  its  nature  and  the 
preceding  considerations  in  regard  to  it.  Consider,  as  is 
represented  in  Figure  2,  a  cylindrical  tube  of  small  diameter 
to  be  placed  in  a  vertical  position  with  its  lower  end  dipping 
into  a  liquid  contained  in  a  larger  vessel,  the  inner  walls  of 
the  tube  having  been  previously  wet  with  the  same  liquid. 
As  a  result  of  the  tendency  of  the  liquid's  surface  to  de- 
crease, the  liquid  rises  in  the  tube  ;  for  thereby  the  free 
surface  of  the  liquid  layer  adhering  to  the  walls  is  caused 
to  disappear  up  to  the  height  to  which  the  liquid  rises.  The 
liquid  column  will  evidently  continue 
to  rise  until  the  constant  surface-force 
which  is  raising  it  is  exactly  compen- 
sated by  the  force  of  gravity  acting 
downward  upon  the  raised  weight  of 
liquid.  Let  the  radius  of  the  tube  be 
r  and  the  height  to  which  the  column 
finally  rises  be  h  ;  let  also,  as  usual,  the 
mass,  volume,  and  density  of  the  raised 
column  of  liquid  be  m,  v,  and  Z>,  and 
the  acceleration  due  to  gravity  be  g. 
Then  the  surface-force  Fs  is  expressed 
by  the  equation  : 


A 
\ 


FIG.  2. 


since  the  line  /  upon  which  the  surface-force  is  acting  is  the 
inner  circumference  (27rr)  of  the  tube.  Moreover,  the  force 
of  gravity  FG  acting  upon  the  raised  liquid  has  the  value  : 


since  TT  i*  h  is  the  volume  of  a  cylinder  of  radius  r  and  height 
h.     Therefore,  when  FS  =  FG, 


Hence,  by  measuring  the  density  of  a  liquid  and  the  height 
to  which  it  rises  in  a  tube  of  known  radius,  its  surface-tension 


KAPH  iil 
CHE*.  BLDG 


tf.  C 


02  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE, 

can  be  determined.  This  height  is  evidently  directly  propor- 
tional to  the  surface-tension  of  the  liquid,  and  inversely 
proportional  to  its  density  and  to  the  radius  of  the  tube.  By 
reason  of  this  last  fact,  it  is  only  in  capillary  tubes  that  con- 
siderable rises  are  observed. 

In  the  general  case,  in  which  a  surface  increases  in 
extent  in  any  number  of  directions,  it  is  still  true  that  dEs  = 
7  ds.  For  then  the  surface-change  ds,  represented  in  Fig- 
ure 3  by  the  area  between  the  outer  and  inner  curves,  can 
be  considered  equal  to  the  sum  of  a  very  large  number  of  in- 
finitesimal surface-changes  of  the  next  higher  order,  each  of 
which  is  the  product  of  a  line  of  infinitesimal  length  dlx  by 
the  distance  dly  through  which  it  is 
displaced;  that  is,  ds=^  (dlx  dly). 
If  now  the  force  acting  on  the  line 
dlx  is  dFs,  the  energy-change  when 
the  line  is  displaced  through  the  dis- 
tance dly  is  evidently  (  dFs  dly  ),  and 
the  total  energy-change  dEs  corre- 
sponding to  the  total  surface-change  ds 
is  the  sum  of  these  elementary  energy- 
F/G  3  changes  ;  that  is,  dEs  =  *Z(  dFs  dly  ). 

Since   the   surface-tension,  the   force 

acting  on  a  line  of  unit-length,  is  known  to  have  the  same 
value  in  all  directions  within  the  surface,  it  is  evidently  true 
for  each  force-element  that  dFs  =  ydtx,  where  7  is  constant. 
It  follows,  therefore,  that  : 


dE,  = 

It  is  found,  experimentally,  that  the  surface-tension  varies 
greatly  with  the  chemical  nature  of  the  liquid,  and  that  it 
decreases  rapidly  with  rise  of  temperature,  but  that  it  is  not 
affected  by  variations  in  the  extent  of  the  surface.  In  view 
of  this  last  fact,  when  the  temperature  is  constant,  the  equa- 
tion dEs  =  7  ds  can  be  integrated,  regarding  7  as  constant, 
with  the  result  : 


ctv 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.  93 


where  J2  and  jt  represent  two  different  extents  of  surface,  and 
ESt  and  Et^  the  corresponding  values  of  the  surface  energy. 

Since  surface-tension  is  defined  to  be  the  surface-force 
exerted  upon  a  line  of  unit-length,  it  is  measured  in  the 
centimeter-gram-second  system  in  dynes  per  centimeter. 

Volume  energy  is  possessed  by  bodies  in  all  three  states 
of  aggregation,  but  the  term  is  applied  to  the  energy  of  solid 
bodies  only  when  it  is  manifested  equally  in  all  directions  ;  in 
other  cases  the  term  elastic  energy  is  used.     Volume  energy 
is  exhibited  in  a  variety  of  phenomena,  and  many  of  these 
are  of  great  physical  and  chemical  impor- 
tance.    As  examples  of  changes  by  which 
external   work   is   done  as    a   result  of   a 
change  of  volume  may  be  mentioned  the 
expansion  of  a  gas  when  the  external  pres- 
sure upon  it  is  sufficiently  reduced,  the  pro- 
duction of  a  volume  of  gas  by  a  chemical 
reaction  against  the  atmospheric  pressure, 
and  the  vaporization  of  a  liquid  under  its 
constant  vapor-pressure.     In  all  such  cases 
the  decrease  of  the  volume  energy  of  the 
body  is   by  definition   equivalent   to    the  maximum  amount 
of  work  that  can   be  obtained  when  the  change  in  volume 
takes  place. 

Changes  in  the  quantity  of  volume  energy  are  most 
readily  calculated  in  the  case  where  the  volume  undergoes  a 
change  in  dimensions  in  one  direction  only.  Suppose,  as  is 
represented  in  Figure  4,  that  a  liquid  or  gaseous  substance 
contained  in  a  cylinder  is  enclosed  by  a  movable  piston  of 
cross-section  s,  and  that  a  force  Ft  is  exerted  upon  this  piston, 
for  example,  by  a  weight  placed  upon  it,  just  sufficient  to  com- 
pensate the  expansive  force  Fv  of  the  body  and  prevent  its 
expansion.  Suppose  now  that  the  external  force  be  reduced 
by  an  infinitesimal  amount  and  that  the  piston  rises  through 
a  distance  dL  The  increase  of  volume  dv  is  then  s  <//,  and 


94  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

the  expansive  force  acting  upon  the  unit  of  surface,  which 
force  is  called  pressure  p,  is  F0  /  s.  The  work  d  W  done  by 
the  body,  or  the  decrease  in  its  volume  energy  ( —  dEv),  is 
therefore  given  by  the  equations  : 


that  is,  it  is  equal  to  the  product  of  the  pressure  into  the 
infinitesimal  increase  of  volume  that  takes  place. 

In  the  general  case  in  which  the  volume  increases  in 
dimensions  in  any  number  of  directions,  it  is  still  true  that 
—  dEv  —p  dv.  This  may  be  demonstrated  by  a  proof  analo- 
gous in  every  respect  to  that  by  which  the  equation  dEs  =  7  ds 
was  shown  to  be  an  entirely  general  one  :  for  in  this  case 
the  volume-change  dv  can  be  considered  equal  to  the  sum  of 
a  large  number  of  infinitesimal  volume-changes  of  the  next 
higher  order,  each  of  which  is  the  product  of  a  surface  of  in- 
finitesimal area  ds  into  the  distance  dl  through  which  it  is 
displaced;  and,  in  the  analogous  calculation  of  the  accom- 
panying energy-change,  the  pressure,  like  the  surface-tension, 
may  be  considered  to  have  the  same  value  in  all  directions. 

The  general  expression  for  the  decrease  in  volume  energy 
Ev  of  a  body  that  undergoes  a  change  of  volume  from  v±  to  z/2> 
or  the  maximum  amount  of  work  W  that  can  be  done  by  it, 
is  therefore : 

If  the  pressure  is  constant  during  the  change  of  volume,  this 
equation  becomes  : 


If,  however,  the  pressure  varies,  as  is  frequently  the  case,  it 
is  evidently  necessary  to  know  the  functional  relation  between 
pressure  and  volume  before  the  integration  can  be  carried  out. 
This  will  be  illustrated  in  the  later  paragraphs  of  this  sec- 
tion, in  which  some  applications  of  the  general  equation  are 
presented. 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.  95 

In  the  centimeter-gram-second  system  pressure  is  ex- 
pressed in  dynes  per  square  centimeter.  Other  units  that 
may  be  employed  for  its  measurement,  the  gram  per  square 
centimeter,  the  height  in  centimeters  of  a  compensating  mer- 
cury column  at  o°,  and  the  atmosphere,  have  been  referred  to 
above.  Since  the  density  of  mercury  at  o°  is  13.596  (almost 
exactly  13.6),  a  column  of  it  one  centimeter  in  height  exerts 
a  pressure  of  13.596  grams  per  square  centimeter;  and  at 
the  sea-level  in  latitude  45°,  where  the  value  of  g  is  980.7, 
this  is  equal  to  a  pressure  of  980.7  X  13.596  or  13333  dynes 
per  square  centimeter.  One  atmosphere,  which  is  by  defini- 
tion equal  to  a  pressure  of  76  cm.  of  mercury,  is  therefore 
equal  to  76  X  13333  or  i  013  300  dynes. 

The  most  important  cases  in  which  a  change  of  volume 
takes  place  under  a  constant  external  pressure  are  those  in 
which  a  pure  liquid  or  solid  substance  vaporizes  at  a  constant 
temperature  against  an  external  pressure  made  substantially 
equal  to  its  vapor  pressure,  and  those  in  which  a  system 
expands  against  the  prevailing  atmospheric  pressure,  owing 
either  to  a  rise  in  its  temperature  or  to  the  occurrence  of  some 
chemical  reaction  within  it ;  or  those  in  which  a  reversal  of 
these  changes  takes  place.  In  such  cases  the  work  done  or 
the  decrease  of  volume  energy  is  found  by  substituting  in  the 
expression  /  (vz  —  Vi)  the  values  of  the  constant  pressure  and 
of  the  final  and  initial  volumes  of  the  system.  The  case 
where  the  change  consists  in  the  production  of  a  gas  from 
liquid  or  solid  substances  deserves,  however,  special  consider- 
ation. If  the  volume  of  the  gas  produced  be  v  and  the  final 
and  initial  volumes  of  the  non-gaseous  part  of  the  system  be 
F2  and  Vl9  the  work  done, 

W=p(v+V,-V,). 

If  the  pressure  is  small,  the  change  in  volume  of  the  non- 
gaseous  part  of  the  system  can  be  neglected  without  giving 
rise  to  any  considerable  error  in  the  calculated  amount  of 
work ;  for  this  volume-change  is  then  very  small  in  compari- 
son with  the  volume  of  the  gas  produced.  In  this  case, 


96  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 


where  N  is  the  number  of  mols  of  gas  produced.  It  is  thus 
seen  that  the  work  done  is  the  same  for  one  mol  of  any  gas, 
is  independent  of  its  pressure  or  volume,  and  is  proportional 
to  the  absolute  temperature. 

In  order  to  calculate  by  this  equation  the  numerical  value 
of  the  work  done,  it  is  necessary  to  know  that  of  the  gas- 
constant  R  expressed  in  ergs,  joules,  or  calories.  Its  value 
was  shown  in  §  24  to  be  0.0820  when  the  pressure  is  expressed 
in  atmospheres  and  the  volume  in  liters  :  since  one  atmos- 
phere is  equal  to  i  013  300  dynes  and  one  liter  to  1000  ccm., 
it  follows  that  : 

R  =  0.0820  X  i  013  300  X  1000  =  8.31  X  io7  ergs  = 

8.31  joules  =  (8.31  74.184)  cal.  =  1.986  cal. 
Therefore,  the  numerical  value  of  the  work  done  is  in  general  : 
^=8.31  X  icPNT  ergs  =  S.$ilVr  joules  =  1.986  NT  cal. 
It  is  convenient  to  remember  that  the  value  of  the  work  done 
is  very  nearly  2  T  calories  for  each  mol  of  gas  produced.  If 
a  gas  condenses  to  a  liquid,  or  is  absorbed  by  a  reacting 
mixture,  the  work  done  has,  of  course,  the  same  numerical 
value,  but  a  negative  sign.  —  The  magnitude  of  the  error 
caused  by  neglecting  the  change  in  volume  of  the  liquid  may 
be  illustrated.  One  gram  of  water  at  100°  has  in  the  liquid 
state  a  volume  of  1.043  ccm.,  and  in  the  form  of  saturated 
vapor  (at  one  atmosphere's  pressure)  a  volume  of  1661  ccm. 
The  former  volume  is  thus  only  0.06  per  cent,  of  the  latter. 
Since  the  volume  of  liquids  is  but  slightly  affected  by  pres- 
sure, while  that  of  gases  is  nearly  inversely  proportional  to 
it,  the  error  in  neglecting  the  former  volume  increases  or 
decreases  nearly  proportionally  with  the  increase  or  decrease 
of  pressure. 

A  very  important  case  of  expansion  in  which  the  pres- 
sure is  variable  is  that  in  which  the  temperature  of  a  quantity 
of  a  gas  remains  constant,  and  its  volume  is  varied  (from  v±  to 
v^).  The  pressure  of  the  gas,  assuming  that  its  value  is  not 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.  97 

too  great,  then  varies  continuously  (from  /x  to  /2)  in  accord- 
ance with  Boyle's  Law,  and  has  for  any  volume  v  the  value 
p  =  NR  T  I  v.  If  now  the  pressure  of  the  gas  is  compen- 
sated by  an  externally  applied  pressure  kept  always  substan- 
tially equal  to  its  own,  the  work  done, 


f  "  —  = 

«/»! 


—  —  ; 

«/»!  V  ?>1 

or,  since  by  Boyle's  Law,  z>2  /*>i=/i//2, 

W=NRT  log-^1. 
A 

Substituting  as  before  the  appropriate  numerical  value  of 
R,  an  expression  for  the  work  done  in  ergs,  joules,  or,  calories 
is  obtained.  For  use  in  numerical  computations  the  expres- 
sion may  be  further  simplified  by  multiplying  the  value  of  R 
by  2.303  and  substituting  ordinary  for  natural  logarithms, 
since  in  general,  Iog10  [  ]  =  2.803  log[  ]•  F°r  example,  in 
order  to  calculate  the  work  done  in  joules  when  one  gram  of 
oxygen  at  20°  and  a  pressure  of  three  atmospheres  expands 
against  an  external  pressure  constantly  kept  substantially 
equal  to  its  own  until  its  pressure  becomes  one  atmosphere, 
we  place  in  the  last  equation,  N—  m  /  M=  i  /  32  ;  R  =  8.3  1  ; 
T=  273  +  20  ;  and  log  (p^  //2)  =  log  3  =  2.303  Iog103  =  2.  303 
X  0.4771  ;  whereby  we  obtain  W=  83.6  joules.  If  in  this 
example  the  words  "  one  liter  of  any  gas  "  were  substituted 
for  "  one  gram  of  oxygen,"  the  problem  would  be  most  readily 
solved  by  putting  NR  T  ==  /  v,  substituting  for  /  its  value 
in  dynes  (3  X  i  013  300)  and  for  v  its  value  in  cubic  centi- 
meters (1000),  multiplying  the  result  by  log  3  as  before,  and 
reducing  to  joules  by  dividing  by  io7. 

One  other  case  of  expansion  at  constant  temperature 
under  a  variable  pressure,  that  of  a  liquid  which  has  been 
compressed  by  the  application  of  external  force,  will  be  briefly 
considered.  Such  a  liquid  is  found  to  exert  a  pressure  which 
is  approximately  proportional  to  the  fractional  decrease  in 


\ 


98  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

its  volume  already  produced ;  that  is,  /  =  —     °         ,  where 

K  F0 

VQ  is  the  volume  of  the  liquid  when  under  no  pressure,  J^its 
volume  when  under  the  pressure  /,  and  K  is  a  quantity,  de- 
fined by  this  equation  and  known  as  the  compression-coefficient, 
which  is  nearly  constant  for  moderate  changes  of  pressure, 
but  variable  with  the  chemical  nature  of  the  substance  and 
the  temperature.  By  substituting  this  value  of  /  in  the  gen- 
eral equation,  dW  —  pdv,  and  integrating,  the  work  corre- 
sponding to  any  definite  volume-change  is  determined.  It  is 
customary,  it  may  be  added,  to  use  the  atmosphere  as  the 
unit  of  pressure  in  stating  the  numerical  value  of  the  com- 
pression-coefficient . 

The  simplest  case  of  elastic  energy,  and  the  only  one 
that  will  be  here  referred  to,  is  that  shown  by  a  bar  of  uniform 
cross-section  which  is  stretched  or  compressed  by  the  applica- 
tion of  forces  at  its  ends ;  for  example,  that  shown  by  a  metal 
wire  fastened  at  its  upper  end  and  supporting  at  its  lower  end 
a  heavy  weight.  In  such  a  case,  which  is  analogous  to  that 
in  which  surface  energy  causes  a  decrease  of  surface  in  only  one 
direction,  the  increase  in  elastic  energy  dEt  is  to  be  regarded 
as  the  product  of  a  force,  called  the  elastic  force  F£y  into  the 
increase  in  length  dl.  That  is,  dEe  =  Fe  dl.  The  elastic  force 
tending  to  cause  the  bar  or  wire  to  become  shorter  is  measured 
by  the  external  force  that  must  be  applied  in  order  to  com- 
pensate this  tendency.  Its  value  is  found  to  be  proportional 
to  the  ratio  of  the  increase  in  length  /  —  /0  that  the  external 
force  has  already  produced  in  the  bar  to  its  original  length  /0, 
inversely  proportional  to  the  cross-section  s  of  the  bar,  and 
variable  with  the  chemical  nature  and  physical  condition  of 
the  substance  composing  it.  The  stated  proportionality  holds 
true,  however,  only  when  the  fractional  increase  in  length  has 
not  exceeded  a  certain  value  known  as  the  elastic  limit. 
These  laws  of  the  elastic  force  developed  under  the  conditions 

stated  are  expressed  by  the  equation,  FE  =  e  s    ~~~  °,  where  e 


GENERAL  PRINCIPLES  R  EL  AIDING   TO  ENERGY.          99 

is  a  quantity,  known  as  the  modulus  of  elasticity,  constant 
with  reference  to  variations  in  the  length  and  cross-section  of 
the  bar,  but  varying  with  its  chemical  composition  and  physical 
condition,  and  fully  defined  by  the  equation  itself.  By  substi- 
tuting this  value  of  FK  in  the  preceding  differential  equation, 
and  integrating,  the  decrease  in  elastic  energy,  or  the  maxi- 
mum work  that  can  be  obtained,  when  a  definite  change  in 
the  length  of  the  bar  takes  place,  is  determined. 

31.  Electricity  and  Magnetism.  Coulomb's  Law. 
Electric  Currents. — Just  as  the  manifestations  of  gravita- 
tion and  kinetic  energies  lead  us  to  assign  to  bodies  con- 
stant inherent  properties  which  express  their  capacity  for 
those  energies,  and  to  conceive  an  entity,  matter,  which  gives 
rise  to  those  properties  (§7),  so  the  phenomena  connected 
with  electrical  and  magnetic  energies  lead  to  the  conclusion 
that  the  bodies  manifesting  them  have  temporarily  acquired 
a  definite  property  upon  which  in  part  depends  the  quantities 
of  these  energies  which  the  bodies  possess,  and  lead  further  to 
the  conception  of  quantities,  called  electricity  and  magnetism, 
which  by  their  temporary  association  with  matter  give  rise 
to  this  property  in  bodies.  Hence,  these  concepts  have  a 
fundamental  relation  to  electrical  and  magnetic  energies 
similar  to  that  which  matter  has  to  gravitation  and  kinetic 
energies.  Therefore,  the  important  principles  relating  to 
them  will  be  first  considered. 

When  a  piece  of  glass  and  a  piece  of  resin  are  rubbed 
or  pressed  together  and  are  then  separated,  they  are  found 
to  have  acquired  the  property  of  attracting  each  other. 
When  another  piece  of  glass  and  another  piece  of  resin 
are  rubbed  and  separated,  it  is  found  that  the  two'  pieces  of 
glass  or  the  two  pieces  of  resin  repel  each  other,  and  that 
either  piece  of  glass  attracts  either  piece  of  resin.  When, 
moreover,  two  other  bodies  of  different  natures  are  rubbed 
and  separated,  it  is  often  found  that  they  attract  each 
other,  and  that  then  one  of  them  always  possesses  the  prop- 
erty of  attracting  a  piece  of  glass  and  repelling  one  of  resin 


100         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

which  have  been  previously  rubbed  together,  and  that  the 
other  body  possesses  the  opposite  property.  The  degree  of 
the  attraction  or  repulsion  is  also  found,  in  general,  to  be 
different  in  the  case  of  the  different  bodies.  It  is  evident, 
then,  that  they  have  acquired  distance  energy  with  respect 
to  one  another,  and  that  each  of  them  separately  has  acquired 
a  property  which  in  part  determines  what  energy  it  possesses 
with  reference  to  any  one  of  the  other  bodies,  and,  further, 
that  this  property  not  only  varies  in  magnitude,  but  is,  in 
the  different  cases,  of  two  opposite  kinds  with  respect  to  the 
direction  of  the  effects  corresponding  to  it.  Hence,  in  order 
to  express  the  existence  and  directional  character  of  this 
property  in  bodies,  it  is  assumed  that  there  are  two  kinds 
of  electricity,  called  positive  and  negative  electricities  ;  and 
those  bodies  which  have  acquired  the  property  of  attracting 
each  other  are  said  to  be  electrified  or  charged  with  the 
opposite  kinds  of  electricity,  and  those  which  repel  each 
other,  with  the  same  kind  of  electricity.  It  is  further  agreed, 
entirely  arbitrarily,  to  designate  the  electricity  upon  glass 
which  has  been  rubbed  with  resin  as  positive  electricity,  and 
that  upon  the  resin  as  negative  electricity,  and  to  designate 
all  other  charges  of  electricity,  in  whatever  manner  produced, 
as  positive  or  negative,  according  as  they  repel  or  attract  the 
charge  upon  the  glass,  or  attract  or  repel  the  charge  upon 
the  resin. 

It  is  agreed,  moreover,  to  measure  quantities  of  elec- 
tricity by  placing  them  proportional  to  the  forces  of  attrac- 
tion or  repulsion  which  indefinitely  small  bodies  charged  with 
them  exert  upon  one  another  when  the  bodies  are  at  a  defi- 
nite distance  and  are  separated  by  a  definite  medium.  And 
correspondingly,  the  C.  G.  S.  electrostatic  unit  of  electricity 
is  defined  to  be  that  quantity  of  electricity  which,  when 
placed  in  air  under  the  normal  conditions  at  a  distance  of 
one  centimeter  from  an  equal  quantity  of  the  same  kind 
of  electricity,  repels  it  with  a  force  of  one  dyne. 

Of  the  important  principles  in  regard  to  quantities  of 


GENERAL  PRINCIPLES  RELATING   TO  ENERGY.         101 

electricity  the  following  one  may  be  first  stated :  The  appear- 
ance of  a  quantity  of  one  kind  of  electricity  at  any  place  is 
always  accompanied  by  the  appearance  of  an  equal  quantity 
of  the  other  kind  of  electricity  at  some  other  place.  If,  as  is 
customary,  quantities  of  positive  and  negative  electricities  are 
considered  to  be  positive  and  negative  numerical  quantities, 
respectively,  this  principle  can  ^also  be  stated  as  follows : 
The  algebraic  sum  of  all  quantities  of  electricity  developed 
by  any  process  whatever  is  equal  to  zero.  This  law  may 
be  demonstrated  by  experiments  like  the  following :  if  two 
disks,  one  of  sealing-wax  and  one  covered  with  flannel,  be 
rubbed  together,  and  either  of  them  be  then  separately  in- 
serted within  a  hollow  metal  vessel  connected  with  a  gold- 
leaf  electroscope,  it  is  found  that  the  gold  leaves  diverge, 
owing  to  their  electrification  by  the  effect  known  as  induc- 
tion ;  if,  however,  both  of  the  disks  be  simultaneously  inserted 
within  the  vessel,  it  is  found  that  no  divergence  takes  place, 
showing  that  the  effects  of  the  opposite  electricities  on  the 
two  disks  just  neutralize  each  other,  and  therefore  that 
the  quantities  of  them  are  equal. 

Another  fundamental  principle  in  regard  to  quantities 
of  electricity  is  that  relating  to  their  attraction  and  repulsion. 
This  may  be  stated  as  follows :  Every  elementary  quantity 
of  electricity  repels  every  other  such  quantity  of  the  same 
kind,  and  attracts  every  other  such  quantity  of  the  opposite 
kind,  with  a  force  varying  directly  as  the  product  of  those 
quantities,  inversely  as  the  square  of  the  distance  between 
them,  and  specifically  with  the  nature  of  the  dielectric. 
The  term  elementary  quantity  of  electricity  is  here  used  to 
express  an  infinitesimal  quantity  of  electricity,  or  any  quan- 
tity of  it  that  can  be  regarded  as  located  at  a  point.  The 
term  dielectric  is  the  one  commonly  employed  to  designate 
the  non-conducting  medium  separating  the  two  quantities  of 
electricity.  The  first  portion  of  this  statement,  which  asserts 
proportionality  between  the  force  of  attraction  or  repulsion 
and  the  quantities  of  electricity,  is  obviously  only  a  converse 


102         GENERAL   PRINCIPLES  OF  PHYSICAL  SCIENCE. 

expression  of  the  above-given  definition  of  these  quantities. 
That  portion  of  it  which  has  reference  to  the  influence  of  dis- 
tance has  been  established  by  experimental  investigations,  and 
is  known  from  its  discoverer  as  Coulomb's  Law. 

These  principles  are  expressed  by  the  equation  : 


where  FE  is  the  force  of  attraction  between  two  quantities 
of  electricity,  &  and  <?„  located  at  points  at  a  distance  /  from 
each  other,  and  K  is  a  positive  quantity,  known  as  the 
dielectric  constant,  which  is  constant  with  reference  to 
variations  of  <2i,  <?2,  and  /,  but  variable  in  a  high  degree  with 
the  nature  of  the  dielectric.  (Thus,  its  value  is  80  times  as 
great  for  water  at  18°  as  it  is  for  air.)  In  this  expression 
the  numerical  values  of  QV  and  <?2  are  to  be  taken  positive  or 
negative  according  as  they  represent  quantities  of  positive 
or  negative  electricity  :  that  of  FE  then  becomes  positive  or 
negative  according  as  those  of  &  and  Q,  have  unlike  or  like 
signs,  a  negative  value  of  FE  evidently  signifying  a  repulsion. 
The  increase  in  distance  energy  dEt  accompanying  an 
increase  in  distance  dl  between  two  elementary  quantities 
of  electricity  is  therefore  expressed  by  the  equations  : 


The  close  analogy  between  these  expressions  and  those 
for  gravitational  force  and  energy  (§  29)  is  apparent.  The 
quantities  <?„  <22,  evidently  have  Ho  electrical  distance  energy 
on  the  one  hand,  and  to  the  concept  of  electricity  on  the 
other,  a  relation  entirely  similar  to  that  which  the  quantities 
M'I,  m'z,  have  to  gravitation  energy  and  to  the  concept  of 
matter. 

Coulomb's  Law  was  originally  derived  and  verified  by 
measuring  the  forces  of  attraction  and  repulsion  between 
two  small  electrified  bodies  by  means  of  the  so-called  torsion- 
balance,  in  which  a  gilded  pith-ball,  attached  at  the  end 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         103 

of  a  light  horizontal  arm  that  is  suspended  at  its  center 
by  a  vertical  wire,  is  electrified  and  placed  at  varying  dis- 
tances from  a  small  electrified  metal  sphere  fixed  in  position ; 
the  relative  values  of  the  electrical  force  being  measured  by 
noting  the  angle  through  which  the  top  of  the  wire  must  be 
turned,  in  order  that  its  torsion  may  just  compensate  the 
tendency  of  the  two  bodies  to  approach  or  recede  from  each 
other.  The  attraction  or  repulsion  of  electrified  bodies  whose 
dimensions  are  considerable  in  comparison  with  the  distance 
between  them  cannot,  however,  be  so  simply  expressed,  not 
only  because  the  charges,  not  being  located  at  definite  points, 
are  not  at  any  one  definite  distance  from  each  other,  but  also 
because,  by  reason  of  the  phenomenon  of  electrical  induction, 
the  charges  mutually  influence  the  distribution  of  each  other 
upon  the  surfaces  of  the  bodies,  so  that  the  elementary  quan- 
tities of  electricity  located  at  definite  points  are  not  the  same 
when  the  bodies  are  near  each  other  as  they  are  when  the 
bodies  are  so  distant  that  no  appreciable  inductive  effect  is 
exerted. 

The  above-stated  law  relating  to  the  attraction  and 
repulsion  of  elementary  quantities  of  electricity  also  deter- 
mines the  distribution  of  electricity  upon  bodies ;  thus,  the 
principles  that  it  resides  entirely  upon  their  surfaces  and  in 
greatest  quantity  upon  those  parts  where  the  curvature  of  the 
surface  is  greatest,  are  consequences  of  it.  Conversely, 
from  the  distribution  as  experimentally  determined  the  law 
itself  has  been  mathematically  derived;  and  it  is  in  this 
way  that  its  exactness  has  been  most  rigidly  demonstrated. 

In  closing  the  consideration  of  statical  charges  of  elec- 
tricity, a  few  terms  in  common  use  may  be  defined.  The 
quantity  of  electricity  on  the  unit  of  surface  is  called  the 
surface-density  of  the  charge.  The  space  surrounding  an 
electric  charge  and  in  such  proximity  to  it  that  an  appreciable 
effect  would  be  exerted  upon  an  electrified  body  that  might 
be  placed  there,  is  called  an  electric  field.  The  force  that 
would  be  exerted  at  any  point  of  an  electric  field  upon  a  unit- 


104         GENERAL   PRINCIPLES  OF  PHYSICAL  SCIENCE. 

charge  of  positive  electricity  considered  to  be  placed  there 
without  itself  influencing  the  field,  is  called  the  intensity  or 
.strength  of  field  at  that  point.  The  lines  in  an  electric  field 
which  represent  the  direction  in  which  such  a  charge  would 
tend  to  move  are  called  lines  of  force.  By  a  pure  convention 
it  is  agreed  to  consider  to  be  drawn  through  any  section 
of  the  field  of  unit-area  that  is  at  right  angles  to  the  lines  of 
force  a  number  of  lines  equal  to  the  intensity  of  field  in  that 
section ;  and  intensity  of  field  is  often  numerically  expressed 
by  stating  the  number  of  lines  of  force  per  square  centimeter, 
in  correspondence  with  this  convention. 

Some  important  principles  relating  to  magnetism  will  be 
next  presented.  When  two  bars  of  steel  are  separately  sub- 
jected to  certain  influences — for  example,  when  they  are 
systematically  rubbed  with  a  lode-stone  or  placed  within  a 
helical  coil  of  wire  through  which  an  electric  current  is  pass- 
ing —  the  corresponding  ends  of  the  two  bars  are  found  to 
repel  each  other  and  the  opposite  ends  to  attract  each  other 
with  forces  varying  within  certain  limits  with  the  influence  to 
which  they  have  been  subjected.  Such  steel  bars,  and  other 
bodies  whose  opposite  parts  possess  the  property  of  attracting 
and  repelling  the  ends  of  such  bars,  are  said  to  be  magnet- 
ized ;  and  it  is  conceived  that  upon  the  ends  are  present 
quantities  of  two  opposite  kinds  of  magnetism.  The  points 
at  which  the  resultants  of  the  attracting  and  repelling  forces 
.act,  and  at  which  the  magnetism  may  be  assumed  to  be 
located,  are  called  the  magnetic  poles.  When  a  magnetized 
bar  or  needle  is  suspended  at  its  middle  point  so  as  to  swing 
freely  in  a  horizontal  plane,  it  is  found  that  it  sets  so  as  to 
point  nearly  north  and  south.  That  pole  that  turns  towards 
the  north  is  called  the  north  pole  of  the  magnet  ;  that  towards 
the  south,  the  south  pole  :  and  the  magnetism  in  a  north  pole 
is  called  north  or  positive  magnetism  ;  that  in  a  south  pole, 
south  or  negative  magnetism. 

The  principles  stated  above  in  regard  to  the  measure- 
ment and  the  attraction  and  repulsion  of  quantities  of  elec- 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         105 

tricity,  and  in  regard  to  the  simultaneous  development  of 
opposite  kinds  of  it,  all  apply  to  quantities  of  magnetism ;  it 
being  only  necessary  in  the  statements  of  them  to  substitute 
the  word  magnetism  for  the  word  electricity.  To  the  state- 
ment of  the  last-mentioned  principle  may  be  added  the  limita- 
tion that  the  equal  quantities  of  the  two  kinds  of  magnetism 
are  always  developed  in  different  parts  of  the  same  continu- 
ous body ;  that  is,  no  body  ever  contains  an  excess  of  either 
kind  of  magnetism :  thus,  in  this  respect,  magnetism  differs 
from  electricity.  The  absolute  unit  of  magnetism,  defined  in 
complete  analogy  with  the  unit  of  electricity,  has  no  specific 
name,  but  is  known  as  the  C.  G.  S.  unit  of  magnetism.  The 
constant  determined  by  the  nature  of  the  intervening  medium, 
occurring  in  the  expression  of  Coulomb's  Law,  is,  in  the  case 
of  magnetism,  known  as  the  magnetic  permeability  ;  its  value 
for  air  under  the  normal  conditions,  like  that  of  the  dielectric 
constant,  is  by  definition  unity.  The  terms  field,  intensity 
or  strength  of  field,  and  lines  of  force  are  used  in  the  same 
sense  in  connection  with  magnetism  as  in  connection  with 
electricity ;  the  word  magnetic  or  electric  being  added  when 
it  is  necessary  to  discriminate.  One  other  term  in  very 
common  use  may  also  be  here  defined ;  this  is  the  term 
strength  of  pole,  which  is  used  as  the  equivalent  of  the  ex- 
pression quantity  of  magnetism  in  the  pole. 

Although  electricity  and  magnetism  have  many  charac- 
teristics in  common,  it  is  to  be  noted  that  stationary  charges 
of  electricity  exert  no  attraction  or  repulsion  on  magnetic 
poles.  They  differ  from  one  another  also  in  that  electricity 
is  capable  of  flowing  under  suitable  conditions  from  one  body 
to  another,  while  magnetism  is  not.  This  flow  of  electricity 
will  be  next  considered. 

If  two  electrified  bodies  are*  connected  by  a  wire,  in 
general  a  momentary  flow  of  electricity  takes  place  from  one 
body  to  the  other,  as  is  shown  by  the  redistribution  of  the 
electricity  upon  the  two  bodies.  If  new  quantities  of  elec- 
tricity are  continuously  imparted  to  the  one  body  and  removed 


106         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

from  the  other  body,  for  example,  by  connecting  them  with 
the  terminals  of  an  electrical  machine,  or,  better,  with  those 
of  a  voltaic  cell,  the  connecting  wire  acquires  certain  new 
properties,  and  is  said  to  be  traversed  by  a  current  of  elec- 
tricity or  an  electric  current.  One  of  these  properties  of  the 
wire  is  that  of  exerting  upon  a  magnetic  pole  a  force  tending 
to  move  it  in  a  direction  at  right  angles  to  the  plane  embrac- 
ing the  wire  and  the  pole  ;  and  it  is  agreed  to  measure  electric 
currents  by  placing  them  proportional  to  this  force  under 
specified  conditions.  Namely,  the  C.  G.  S.  electro-magnetic 
unit  of  current  is  the  current  which,  flowing  in  a  conductor 
of  the  form  of  a  circular  arc  one  centimeter  long,  with  a 
radius  of  one  centimeter,  acts  with  a  force  of  one  dyne 
on  a  magnetic  pole  of  unit-strength  placed  at  the  center 
of  the  circle.  The  C.  G.  S.  electro-magnetic  unit  of  electric- 
ity is  then  defined  to  be  the  quantity  of  it  which  passes 
between  two  points  in  one  second  when  the  C.  G.  S.  electro- 
magnetic unit  of  current  is  flowing  between  them.  There- 
fore, in  general,  designating  the  current  by  /,  and  the  time 
during  which  it  flows  by  T,  the  total  quantity  of  electricity  <? 
which  passes  in  that  time  is  given  by  the  equation,  Q  =  IT. 
It  should  be  added  that  it  is  agreed  to  understand  by  the 
direction  of  the  current  the  direction  in  which  positive  elec- 
tricity, as  electrostatically  defined,  is  flowing. 

The  general  law  governing  the  force  F  exerted  by  a 
length  /  of  a  circular  arc,  of  radius  r,  through  which  a  cur- 
rent of  strength  /  is  flowing,  upon  a  magnetic  pole  of  strength 
M  placed  in  air  at  the  center  of  the  circle,  is  expressed  by  the 
equation  :  F—  MI  1 1  r*.  The  direction  in  the  line  perpendic- 
ular to  the  plane  of  the  circuit  and  magnetic  pole  in  which  the 
force  acts,  is  determined  both  by  the  direction  of  the  current 
and  the  sign  of  the  magnetism  in  the  pole,  in  a  manner  that 
need  not  be  here  specified.  This  action  exerted  on  quantities 
of  magnetism  by  electric  currents  shows  that  a  magnetic  field 
is  produced  in  the  neighborhood  of  the  current.  Thus,  it  is 
seen  that,  though,  as  stated  above,  there  are  no  effects  exerted 


GENERAL  PRINCIPLES  RELATING   TO  ENERGY.         107 

between  magnetic  poles  and  stationary  electric  charges,  yet 
there  are  important  relations  between  the  former  and  electric 
currents. 

The  ratio  of  the  electromagnetic  to  the  electrostatic 
unit  of  electricity  is  evidently  one  of  the  most  fundamental 
constants  connected  with  electrical  phenomena,  since  it  makes 
it  possible  to  bring  the  effects  of  statical  charges  into  quanti- 
tative relations  with  those  of  electric  currents.  Its  numerical 
value  has  been  found  to  be  3.00  X  io10  by  means  of  direct 
experimental  comparisons  of  these  two  kinds  of  effects.  The 
very  high  value  of  this  constant  shows  that  quantities  of 
electricity  which  in  the  form  of  statical  charges  exert  a  con- 
siderable force  upon  one  another  give  rise  to  only  an  ex- 
tremely slight  electromagnetic  force  when  in  the  form  of  a 
current ;  moreover,  it  is  true  in  general  that  the  quantities 
of  electricity  concerned  in  most  electrostatic  phenomena  are 
almost  inappreciable  in  comparison  with  those  ordinarily 
involved  in  the  case  of  electric  currents.  Incidentally,  it 
may  be  mentioned  that  the  value  of  the  ratio  of  the  two 
units  has  been  found  to  be  identical,  within  the  experimental 
error,  with  the  velocity  of  light  —  a  remarkable  relation  of 
great  theoretical  significance. 

32.  The  Factors  of  Electrical  Energy.  Ohm's  Law 
and  Joule's  Law.  —  In  the  preceding  section  an  expression 
was  given  for  the  change  in  the  electrical  energy  of  two 
quantities  of  electricity  corresponding  to  a  displacement  of 
one  of  them  in  the  direction  in  which  the  electric  force  is 
acting.  This  is  obviously  only  a  special  case.  In  order  to 
express  in  a  general  manner  the  change  in  electrical  energy, 
—  that  produced  by  the  displacement  of  a  quantity  of  elec- 
tricity in  any  direction  whatever  in  any  electric  field  whatever, 
—  the  change  in  energy  is  considered  to  be  the  product  of 
two  factors :  one  of  these  is  quantity  of  electricity,  to  which 
under  otherwise  constant  conditions  electric  force  and  energy 
have  already  been  seen  to  be  proportional ;  and  the  other  is 
difference  of  potential,  which  is  defined  to  be  that  factor  with 


108         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

which  the  quantity  of  electricity  must  be  multiplied  in  order 
to  produce  the  energy.  That  is,  if  a  quantity  of  electricity 
Q  when  transferred  from  one  point  to  another  in  an  electric 
field  suffers  a  change  dE  in  its  electrical  energy,  the  differ- 
ence of  potential  dv  between  the  two  points  is  defined  by  the 
equation,  dv  =  dE  /  Q  .  This  method  of  defining  the  factors 
of  electrical  energy  is  evidently  an  application  of  the  general 
equation,  c  =  dE  /  di,  to  a  case  in  which  c  and  dE  are  first 
independently  determined. 

In  accordance  with  this  equation,  the  C.  G.  S.  unit  of 
potential-difference  is  the  potential-difference  that  exists  be- 
tween two  points  when,  upon  transferring  from  one  of  them 
to  the  other  a  unit  of  electricity,  its  energy  changes  by  one 
erg.  Corresponding  to  the  electrostatic  and  electromagnetic 
units  of  electricity,  there  are,  of  course,  two  units  of  potential- 
difference.  It  is  also  agreed  to  adopt  as  the  zero  of  potential 
the  value  of  it  at  a  point  infinitely  distant  from  the  charges 
producing  the  electric  field,  which  is  practically  identical  with 
the  value  at  any  point  in  electrical  connection  with  the  body 
of  the  earth.  By  integrating  the  equation,  dv~  dE  /  Q,  under 
this  assumption  that  the  potential  is  zero  when  the  distance 
is  infinity,  we  get  for  the  absolute  value  of  the  potential  v  at 
any  point,  v  =  (E  —  E^)  /  Q,  where  E  and  EM  are  the  elec- 
trical energies  at  that  point  and  at  an  infinite  distance,  re- 
spectively. By  putting  Q  =  + 1  in  this  equation,  we  find  that 
the  potential  at  any  point  of  an  electric  field  is  mathematic- 
ally equivalent  to  the  increase  in  the  electrical  energy  of  a 
unit-quantity  of  positive  electricity  when  this  is  transferred 
from  an  infinite  distance  to  the  point  in  question.  Its  numer- 
ical value  is  evidently  positive  when  the  energy-increase  is 
positive,  and  negative  when  it  is  negative.  This  energy- 
increase,  and  therefore  also  the  potential,  is  obviously  equal 
to  the  external  work  that  must  be  expended  in  transferring 
the  unit-quantity  of  positive  electricity  from  an  infinite  dis- 
tance to  the  point  in  question. 

Since   an  energy-increase  accompanying  a  decrease  of 


GENERAL   PRINCIPLES  RELATING    TO  ENERGY.         119 

distance  corresponds  to  a  force  of  repulsion,  it  is  clear  that, 
by  reason  of  the  definitions  adopted,  positive  electricity  tends 
to  flow  from  a  place  at  higher  to  one  at  lower  potential. 
Negative  electricity,  on  the  other  hand,  evidently  will  tend 
to  flow  from  a  place  at  lower  to  one  at  higher  potential. 
Potential  is  therefore  the  intensity-factor  of  electrical  energy, 
and  it  has  the  peculiarity  that  a  definite  difference  in  its  value 
causes  the  energy  to  transfer  itself  in  one  direction  or  the 
opposite  one,  according  as  the  electricity  involved  is  positive 
or  negative.  Since,  however,  almost  all  of  the  effects  pro- 
duced by  the  flow  of  positive  electricity  in  the  one  direction 
are  identical  with  those  produced  by  the  flow  of  negative 
electricity  in  the  other,  it  is  in  most  cases  impossible  to 
determine  to  what  extent  the  two  separate  flows  are  taking 
place,  and  it  is  customary,  for  convenience  sake,  to  consider 
that  an  electric  current  is  wholly  one  of  positive  electricity, 
its  magnitude  as  well  as  its  direction  being  determined 
from  its  effects  by  attributing  these  entirely  to  this  kind  of 
electricity. 

The  relation  of  quantity  of  electricity  and  difference  of 
potential  to  electrical  energy  is  made  clearer  by  considering 
the  analogy  between  it  and  the  relation  of  quantity  of  water 
and  difference  of  level  to  gravitation  energy.  Water  situated 
at  any  level  above  that  of  the  sea  possesses  a  quantity  of 
gravitation  energy  which  is  greater,  the  greater  the  quantity 
of  water  and  the  greater  the  difference  between  the  two 
levels,  just  as  the  quantity  of  electrical  energy  is  greater,  the 
greater  the  quantity  of  electricity  and  the  greater  the  differ- 
ence of  potential.  Moreover,  in  order  that  there  may  be  a 
tendency  for  water  to  flow  from  one  place  to  another,  there 
must  be  a  difference  of  level  between  the  two  places,  just  as 
there  must  be  a  difference  of  potential  at  two  places  in  order 
that  there  may  be  a  tendency  for  electricity  to  flow  between 
them. 

In  place  of  the  expression  difference  of  potential,  it  is 
very  common,  when  electric  currents  are  under  consideration, 


110         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

to  use  the  more  general  term  electromotive  force  (^),  which 
signifies  any  cause  whatever  that  tends  to  produce  a  flow  of 
electricity. 

Magnetic  energy  is  resolved  into  the  factors  quantity  of 
magnetism  and  magnetic  potential,  which  have  a  significance 
entirely  similar  to  that  of  the  factors  of  electrical  energy. 
Quantity  of  magnetism  was  considered  in  the  preceding  sec- 
tion. Magnetic  potential  is  defined  with  reference  to  the 
transfer  of  a  quantity  of  magnetism  in  exactly  the  same 
manner  as  electric  potential  has  been  defined  with  reference 
to  that  of  a  quantity  of  electricity. 

Certain  important  principles  relating  to  electric  currents 
will  be  next  considered.  In  order  that  a  current  may  flow 
between  two  points,  it  is  necessary,  not  only  that  they  be  at 
different  potentials,  but  also  that  they  be  connected  with  each 
other  by  a  suitable  medium.  A  medium  which  allows  an 
appreciable  current  of  electricity  to  flow  through  it  is  called 
an  electrical  conductor ;  one  which  does  not,  a  non-conductor 
or  insulator.  In  regard  to  the  strength  of  the  current  the 
following  general  principle,  known  as  Ohms  Law,  has  been 
derived  and  thoroughly  confirmed  by  experiment:  When  a 
steady  current  flows  between  two  points  without  producing  any 
form  of  work,  its  strength  (/)  varies  directly  as  the  difference 
(E)  of  the  potentials  of  the  points,  and  specifically  with  the 
character  of  the  conductor  connecting  them.  That  is,  /  =  EJ  R, 
where  R  is  a  quantity,  called  the  resistance  of  the  conductor, 
constant  with  reference  to  variations  of  /  and  E,  but  varying 
with  the  form,  chemical  nature,  and  physical  condition  of  the 
conductor.  Since  by  this  equation  resistance  is  defined  to  be 
equal  to  the  ratio  (E/I)  of  potential-difference  to  current 
when  no  work  is  produced  by  it,  the  C.  G.  S.  unit  of  resist- 
ance is  also  determined :  it  is  equal  to  the  resistance  of  a  con- 
ductor through  which  a  C.  G.  S.  unit  of  current  flows  when 
there  is  a  difference  of  a  C.  G.  S.  unit  of  potential  at  its  ends. 
The  resistance  of  a  homogeneous  conductor  of  uniform 
dimensions  is  found  to  be  proportional  to  its  length  (/)  and 


GENERAL   PRINCIPLES  RELATING    TO  ENERGY.         Ill 

inversely  proportional  to  its  cross-section  (s) ;  that  is,  R  =  R_  //  s, 
where  R  is  a  quantity,  designated  the  specific  resistance  or  the 
resistivity  of  the  substance,  which  is  defined  by  this  equation, 
and  which  is,  therefore,  in  terms  of  the  units  commonly  em- 
ployed, the  resistance  of  a  prism  of  it  one  centimeter  in  length 
and  one  square  centimeter  in  cross-section.  Its  value  varies 
not  only  with  the  chemical  nature  of  the  substance,  but  also 
with  physical  conditions,  especially  with  the  temperature.  It 
is  often  more  convenient  to  employ  in  place  of  the  resistance 
of  a  conductor  the  reciprocal  of  its  value,  which  is  called 
the  conductance  (L)  ;  this  is  evidently  equal  to  the  ratio  of  the 
current  to  the  potential-difference  ;  that  is,  L  =  1  /  R  —  /  /  E. 
The  conductance  of  a  homogeneous  conductor  of  uniform 
dimensions  is  directly  proportional  to  its  cross-section  and 
inversely  proportional  to  its  length  ;  that  is,  L  =  L  s  /  /,  where 
L  is  the  specific  conductance  or  the  conductivity  of  the  sub- 
stance, or  the  conductance  of  a  prism  of  it  one  centimeter  in 
length  and  one  square  centimeter  in  cross-section. 

When  work  of  any  kind  is  produced  by  the  passage  of  a 
current  (for  example,  when  mechanical  energy  is  produced  by 
a  motor,  when  electrical  energy  is  produced  in  another  con- 
ductor by  a  transformer,  or  when  chemical  energy  is  produced 
by  a  chemical  decomposition),  a  counter  or  back  electro- 
motive force  is  developed  in  the  circuit,  and  the  current- 
strength  is  correspondingly  reduced.  The  statement  of  Ohm's 
Law  applicable  to  this  general  case  is  that  the  current-strength 
is  equal  to  the  ratio  of  the  algebraic  sum  of  all  the  electro- 
motive forces  to  the  sum  of  all  the  resistances  in  the  circuit. 
That  is,  /  — 2fr  /2*.  For  example,  suppose  that  a  storage 
battery,  whose  internal  resistance  is  R»  and  which  produces 
an  electromotive  force  slf  is  placed  in  a  circuit  with  a  motor 
of  resistance  xz  and  an  electrolytic  cell  of  resistance  xs,  the 
whole  being  connected  by  metallic  conductors  of  resistance 
xt ;  suppose  further  that  the  activity  of  the  motor  gives  rise 
to  a  back  electromotive  force  *2,  and  that  the  chemical  decom- 
position in  the  cell  produces  a  back  electromotive  force  ^8 : 


112         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIEA'CE. 

the  strength  /  of  the  current  flowing  through  the  circuit  is 
then  given  by  the  equation,  /  =      £l  ~  Ef      £*  __ 

*1  +  *2  +  *3  +  *i 

When  a  quantity  of  electricity  Q  flows  between  two 
points  and  thereby  undergoes  a  fall  of  potential  E>  its  electrical 
energy  decreases  by  an  amount  equal  to  the  product  E  Q,  and 
must  appear  as  some  other  form  of  energy,  commonly  as  heat, 
chemical  energy,  or  mechanical  energy,  or  else  as  electrical 
energy  in  another  conductor.  If  a  quantity  of  heat  Q  is  alone 
produced,  as  is  the  case  when  the  two  points  are  connected 
by  an  ordinary  metallic  conductor,  we  get  by  the  substitution 
of  the  value  of  E  from  the  equation  expressing  Ohm's  Law  : 


This  equation  states  that  the  heat  developed  in  a  conductor  by 
the  passage  of  a  current  which  produces  no  work  is  equal  to 
the  square  of  the  current-strength  multiplied  by  the  time  dur- 
ing which  the  current  passes  and  by  the  resistance  of  the 
conductor.  This  principle  is  known  as  Joule  s  Law.  It  has 
received  independent  confirmation  by  direct  experiments. 
When  the  current,  resistance,  and  time  are  expressed  in 
C.  G.  S.  units,  the  heat-effect  is,  of  course,  expressed  in  ergs. 
When  mechanical  or  electrical  work  is  produced  by  a 
current  (as  in  an  electric  motor  or  in  a  transformer),  it  is  found 
experimentally  to  be  still  true  that  a  heat  -effect  equal  to  I*R  T 
is  also  produced,  so  that  Joule's  Law  is  applicable  to  this  case 
also.  When  chemical  energy,  however,  is  produced  by  an 
electric  current  (as  by  the  decomposition  of  a  chemical  sub- 
stance), the  total  heat-effect  is  not  in  general  equal  to  PRT  ; 
there  are,  nevertheless,  good  reasons  for  regarding  it  as  the 
algebraic  sum  of  two  separate  heat-effects,  one  of  which  is 
equal  to  /2  R  T  ;  this  is  called  the  Joule  Heat-Effect  ',  to  distin- 
guish it  from  the  total  heat-effect.  —  Consider,  for  example, 
the  application  of  these  principles  to  the  circuit  described 
above,  containing  a  storage  battery,  a  motor,  and  an  electro- 
lytic cell.  If  the  current  /  flo.ws  for  a  time  T,  the  storage 


GENERAL   PRINCIPLES  RELATING   TO  ENERGY.         113 

cell  yields  a  quantity  of  electrical  energy  E±  i  r.  Of  this 
energy,  by  reason  of  the  resistances,  the  quantities  /2  xl  r, 
/2  /?2  r,  /2  KZ  T  and  /2  *4  r  are  converted  into  heat  in  the  storage 
cell  itself,  the  coils  of  the  motor,  the  electrolytic  cell,  and  the 
metallic  conductors,  respectively.  The  motor  takes,  in  addi- 
tion, a  quantity  of  energy  E2  /  r  which  is  convertible  into 
work.  The  remainder  of  the  energy  ES  /  r  is  consumed  in 
the  electrolytic  cell,  appearing  there  in  the  form  of  chemical 
energy  (that  possessed  by  the  products  of  the  decomposition) 
and  sometimes  partly  in  the  form  of  heat. 

In  closing  the  consideration  of  electrical  energy  and  the 
relations  of  its  factors,  a  few  words  must  be  added  in  regard 
to  electrical  units.  Since  many  of  the  C.  G.  S.  units,  both 
in  the  electrostatic  and  electromagnetic  systems,  are  of  an 
extremely  inconvenient  magnitude  for  the  expression  of  the 
electrical  quantities  involved  in  ordinary  work  with  electric 
currents,  a  so-called  practical  system  of  electrical  units  has 
been  devised,  and  this  system,  or  one  substantially  identical 
with  it,  is  used  almost  exclusively  both  in  electrical  science 
and  electrical  engineering.  All  the  units  in  this  system  are 
exact  decimal  multiples  or  submultiples  of  the  corresponding 
C.  G.  S.  electromagnetic  units.  They  are  designated  and 
denned  as  follows  :  The  unit  of  current  is  called  the  ampere 
and  is  denned  to  be  one-tenth  of  the  C.  G.  S.  electromagnetic 
unit  of  current,  which  was  defined  in  the  preceding  section. 
The  unit-quantity  of  electricity  is  the  coulomb,  which  is  the 
quantity  flowing  per  second  when  the  current  is  one  ampere. 
The  unit  of  potential  is  the  volt,  which  is  equal  to  the  poten- 
tial-difference between  two  points  when  a  change  of  one  joule 
(equal  to  io7  ergs)  in  the  electrical  energy  attends  the  pas- 
sage of  one  coulomb  of  electricity  between  them :  it  is 
evidently  equal  to  io8  C.  G.  S.  electromagnetic  units  of 
potential.  The  unit  of  resistance  is  the  ohm,  which  is  the 
resistance  of  a  conductor  through  which  one  ampere  flows 
when  the  potential-difference  is  one  volt :  it  is  equal  to 
io9  C.  G.  S.  units  of  resistance.  The  unit  of  conductance  is 


114         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

the  reciprocal  ohm,  which  is  the  conductance  of  a  conductor 
whose  resistance  is  one  ohm.  Careful  experimental  determi- 
nations have  shown  that  the  ohm  is  substantially  equivalent 
to  the  resistance  at  o°  of  a  mercury  column  106.3  centi- 
meters in  length  and  of  a  uniform  cross-section  one  square 
millimeter  in  area.  It  should  be  added  that,  in  order  to 
facilitate  the  expression  of  electrical  quantities  in  terms  of 
these  units,  certain  substantial  equivalents  of  them,  more 
readily  realizable  experimentally,  like  the  equivalent  of  the 
ohm  just  mentioned,  have  been  recently  agreed  upon  by 
congresses  of  electrical  engineers  and  various  governments, 
and  have  been  adopted  as  new  definitions  of  the  electrical 
units.  These  are  designated  the  international  units  (inter- 
national ohm,  international  ampere,  etc.)  in  order  to  distin- 
guish them  from  the  true  units  (true  ohm,  true  ampere,  etc.) 
which  were  just  defined.  It  is  highly  improbable  that  the 
values  of  the  international  and  true  units  differ  in  any  case 
by  as  much  as  o.i  per  cent. ;  so  that  for  most  purposes  they 
can  be  regarded  as  identical. 

33.  Faraday's  Law  of  Electrolytic  Conduction. — 
Conductors  are  divided  into  two  classes  with  reference  to  the 
changes  that  are  produced  in  them  by  the  passage  of  electric 
currents.  Those  which  undergo  no  changes  except  such  as 
are  produced  by  a  rise  in  temperature  are  called  metallic  con- 
ductors. Those  in  which  the  passage  of  a  current  is  attended 
by  a  chemical  change  are  called  electrolytes.  Bodies  com- 
posed of  the  various  metals  and  of  graphite  are  examples  of 
good  metallic  conductors.  Aqueous  solutions  of  salts,  bases, 
and  acids,  and  melted  salts  at  high  temperatures,  are  the  most 
important  classes  of  well-conducting  electrolytes.  The  most 
obvious  chemical  changes  attending  the  passage  of  a  current 
through  an  electrolyte  are  those  that  take  place  at  the  sur- 
faces of  the  metallic  conductors  where  the  current  enters  and 
leaves  the  electrolyte.  The  production  of  such  chemical 
changes  by  a  current  from  an  external  source  is  called 
electrolysis.  The  occurrence  of  such  changes,  when  they 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         115 

themselves  give  rise  to  an  electric  current,  is  called  voltaic 
action.  Those  portions  of  the  metallic  conductors  that  are 
in  contact  with  the  electrolyte  are  called  the  electrodes :  the 
one  at  which  the  current  leaves  the  electrolyte,  or  the  one  to 
which  the  positive  electricity  flows  through  the  electrolyte,  is 
designated  the  cathode  or  negative  electrode ;  the  other,  the 
anode  or  positive  electrode. 

The  chemical  changes  at  the  electrodes  accompanying 
the  electrolysis  of  aqueous  solutions  of  salts,  bases,  and  acids 
consist,  in  the  simplest  case,  in  the  precipitation  of  the  metal 
of  the  salt  or  base  or  of  the  hydrogen  of  the  acid,  or  of 
hydrogen  from  the  water,  upon  the  cathode,  and  in  the  pre- 
cipitation of  the  halogen  or  acid  radical  of  the  salt  or  acid  or 
of  the  oxygen  of  the  base,  or  oxygen  from  the  water,  upon 
the  anode :  thus,  when  a  concentrated  solution  of  copper 
chloride  is  electrolyzed  between  carbon  electrodes,  copper  is 
deposited  on  the  cathode  and  chlorine  is  liberated  at  the 
anode  ;  and,  when  a  solution  of  sodium  sulphate,  one  of 
sulphuric  acid,  or  one  of  sodium  hydroxide,  is  electrolyzed 
between  platinum  electrodes,  hydrogen  is  liberated  at  the 
cathode  and  oxygen  at  the  anode.  In  many  cases,  however, 
the  chemical  changes  are  more  complicated,  owing  to  the 
occurrence  of  secondary  reactions  between  the  primary  prod- 
ucts of  the  electrolysis  and  the  substances  composing  the 
electrodes  or  the  electrolyte.  Thus,  when  silver  nitrate  solu- 
tion is  electrolyzed  between  silver  electrodes,  the  NO3-radical 
or  oxygen,  instead  of  being  precipitated  at  the  anode,  attacks 
the  silver  of  which  it  is  composed  and  causes  an  equivalent 
quantity  of  it  to  pass  into  solution  in  the  form  of  silver 
nitrate ;  and,  when  nitric  acid  is  electrolyzed  between  plati- 
num electrodes,  the  hydrogen,  instead  of  being  set  free  in  the 
form  of  gas  at  the  cathode,  reduces  the  nitric  acid,  and  is 
itself  oxidized  to  water.  Furthermore,  whenever  the  electroly- 
sis of  a  neutral  salt  (like  that  of  sodium  sulphate,  for  example) 
is  attended  by  the  liberation  of  hydrogen  at  the  cathode  and 
oxygen  at  the  anode,  it  is  found  that  a  quantity  of  free  base 


116         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

(sodium  hydroxide)  equivalent  to  the  hydrogen,  and  one  of 
free  acid  (sulphuric  acid)  equivalent  to  the  oxygen,  are  pres- 
ent in  the  portions  of  the  electrolyte  surrounding  the  cathode 
and  anode,  respectively,  so  that,  in  this  case  also,  the  effects 
observed  are  those  that  would  result  if  the  metal  and  acid- 
radical  were  the  primary  products  of  the  electrolysis  and  these 
reacted  secondarily  with  the  water ;  thus,  the  phenomena 
observed  in  the  case  of  the  sodium  sulphate  are  those  that 
would  arise  from  the  liberation  of  sodium  and  the  sulphuric- 
acid  radical  at  the  electrodes,  and  their  action  upon  the  water 
according  to  the  reactions,  2Na  +  H2O  =  2NaOH  +  H2,  and 
2SO4  +  2H20  =  2H2S04  +  02. 

In  the  case  of  voltaic  actions,  the  chemical  changes  are 
of  a  similar  character,  most  commonly  consisting  in  the 
solution  of  the  metal  composing  the  anode  and  in  the  deposi- 
tion of  another  metal  or  hydrogen  on  the  cathode  (the  sepa- 
ration of  free  hydrogen  being,  however,  often  prevented  by 
a  secondary  reaction  between  it  and  the  electrolyte  or  the 
electrode).  Thus,  in  the  Daniell  Cell,  which  consists  of  a 
copper  electrode  in  a  copper  sulphate  solution  and  of  a  zinc 
electrode  in  a  zinc  sulphate  solution,  the  two  solutions  being 
in  contact  and  the  two  electrodes  connected  by  a  metallic 
conductor,  the  zinc  dissolves  and  the  copper  precipitates  ; 
and,  in  the  Grove  Cell,  consisting  of  a  zinc  electrode  in  dilute 
sulphuric  acid  and  a  platinum  electrode  in  strong  nitric  acid, 
zinc  dissolves  at  the  anode,  and  the  hydrogen  primarily  pro- 
duced at  the  cathode  reduces  the  nitric  acid  to  lower  oxides 
of  nitrogen.  The  chemical  changes  involved  in  voltaic  actions 
do  not  differ,  therefore,  essentially  from  those  produced  by 
electrolysis,  except  in  the  respect  that  the  former  tend  to  take 
place  of  themselves,  while  the  latter  require  the  application 
of  electrical  energy  from  an  external  source. 

The  amount  of  chemical  change  that  takes  place  at  the 
electrodes  is  determined  by  the  following  fundamental  prin- 
ciple, which  is  known  from  its  discoverer  as  Faraday  s  Law  : 
The  passage  of  electricity  through  an  electrolyte  is  attended 


GENERAL   PRINCIPLES  RELATING    TO  ENERGY.         117 

at  each  electrode  by  a  chemical  change  involving  a  number  of 
chemical  equivalents  strictly  proportional  to  the  quantity  of 
electricity  passed  through  and  dependent  on  that  alone. 

This  law  is  algebraically  expressed  by  the  equation, 
Q  =  Q_N,  in  which  Q  is  the  quantity  of  electricity  passed 
through,  N  is  the  number  of  chemical  equivalents  involved 
in  the  change  at  each  electrode,  and  £  is  a  constant  with 
respect  to  all  variations  of  the  conditions,  representing  the 
quantity  of  electricity  producing  a  chemical  change  involving 
one  equivalent. 

The  law  requires,  on  account  of  its  generality,  illustra- 
tion by  a  number  of  examples.  Consider  first  the  electrolysis 
of  a  silver  nitrate  solution  between  silver  electrodes  by  the 
passage  through  it  of  a  definite  quantity  of  electricity,  say 
1000  coulombs,  under  variable  conditions,  assuming,  how- 
ever, that  the  conditions  are  always  such  that  the  only  chem- 
ical changes  taking  place  at  the  electrodes  are  the  deposition 
of  silver  upon  the  cathode  and  the  solution  of  silver  from 
the  anode.  Faraday's  Law  then  requires  that  the  quantity  of 
silver  deposited  on  the  cathode  be  the  same,  and  that  the 
quantity  dissolved  from  the  anode  be  the  same,  whether  the 
silver  nitrate  solution  be  concentrated  or  dilute,  whether  it 
be  hot  or  cold,  whether  the  electrodes  be  so  placed  in  it  that 
its  resistance  is  large  or  small,  and  whether  the  potential- 
difference  at  the  electrodes,  and  therefore  the  current- 
strength  and  the  electrical  energy  expended,  be  greater  or 
less ;  that  is,  in  spite  of  all  these  and  other  possible  varia- 
tions of  conditions,  the  law  requires  that  whenever  1000 
coulombs  of  electricity  have  passed  through  the  solution  a 
definite  quantity  (found  to  be  1.117  grams)  of  silver  be  pre- 
cipitated at  the  cathode  and  dissolved  at  the  anode.  Sup- 
pose that  in  a  second  series  of  experiments  10000  coulombs 
be  passed  through  the  solution ;  then  ten  times  as  much 
silver  (11.17  grams)  must  be  deposited  and  dissolved. 
Suppose  next  that,  while  still  passing  10000  coulombs, 
the  solution  be  made  so  dilute  or  the  current-strength  so 


118         GENERAL   PRINCIPLES  OF  PHYSICAL  SCIENCE. 

great  in  proportion  to  the  electrode-surface  that  hydrogen  is 
deposited  with  the  silver  upon  the  cathode  :  the  law  requires 
in  this  case  that  the  sum  of  the  number  of  equivalents  of 
hydrogen  and  silver  be  equal  to  the  number  of  equivalents 
of  silver  deposited  in  the  preceding  experiments  ;  thus,  if  the 
amount  of  silver  now  deposited  be  10.00  grams,  that  is,  1.17 
grams  or  1.17/107.93  or  0.01084  equivalents  less  than  be- 
fore, this  deficit  in  silver  must  be  made  up  by  the  liberation 
of  an  equal  number  of  equivalents  of  hydrogen  gas,  that  is, 
since  its  equivalent  weight  is  1.008,  of  0.01093  grams  of  it. 
Consider  next  that  the  same  current  traverses  in  series  the 
following  solutions  placed  between  platinum  electrodes : 
sulphuric  acid  (H2SO4),  silver  nitrate  (AgNO3),  mercurous 
sulphate  (Hg2SO4),  mercuric  bromide  (HgBr2),  gold  chlo- 
ride (AuCl3),  and  platinic  chloride  (PtCl4)  ;  that  only  a 
single  chemical  change  takes  place  at  the  electrodes  in  each 
solution  —  the  precipitation  of  hydrogen  or  the  corresponding 
metal  at  the  cathode,  and  of  oxygen  or  halogen  at  the 
anode;  and  that  the  current  is  continued  till  1.008  grams  of 
hydrogen,  one  equivalent  weigh^have  been  liberated  from  the 
sulphuric  acid  solution.  It  follows  then  from  Faraday's  Law 
that  one  equivalent  weight  of  each  of  the  metals  and  of 
oxygen  or  halogen  will  also  be  deposited ;  namely,  since  the 
combining  weights  (see  §  16)  corresponding  to  the  above- 
given  formulas  are,  Ag  =  107.93,  Hg  =  200.0,  Au  =  197.3, 
Pt  =  195.2,  O  =  16.00,  Br  =  79.96,  and  Cl  =  35.45,  and  since 
the  equivalent  weights  in  the  different  cases  are  evidently 
represented  by  lAg,  iHg,  |Hg,  lAu,  JPt,  i<3,  iBr,  and  id, 
the  result  of  the  electrolysis  is  the  precipitation  of  i  .008  grams 
of  hydrogen  and  8.00  of  oxygen  from  the  sulphuric  acid,  of 
107.93  of  silver  and  8.00  of  oxygen  from  the  silver  nitrate, 
•of  200 .o  of  mercury  and  8.00  of  oxygen  from  the  mercurous 
sulphate,  of  100.0  of  mercury  and  79.96  of  bromine  from 
the  mercuric  bromide,  of  65.77  of  gold  and  35.45  of  chlorine 
from  the  gold  chloride,  and  of  48.80  of  platinum  and  35.45 
of  chlorine  from  the  platinic  chloride.  Consider  finally  the 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         119- 

application  of  the  law  to  the  voltaic  actions  taking  place  in 
the  Daniell  Cell  when  a  quantity  of  electricity  is  produced 
by  it  which  deposits  107.93  grams  or  one  equivalent  of  silver 
from  a  silver  nitrate  solution  placed  in  the  circuit  in  series 
with  the  cell.  Under  these  circumstances,  according  to 
Faraday's  Law,  one  equivalent  of  zinc,  equal  to  |-Zn  or  32.70 
grams,  will  dissolve  from  the  anode  of  the  cell,  and  one 
equivalent  of  copper,  equal  to  ^Cu  or  31.80  grams,  will  pre- 
cipitate upon  its  cathode. 

Although  numerous  determinations  have  been  made 
which  prove  the  validity  of  Faraday's  Law  as  an  approxima- 
tion, it  has  been  found  very  difficult  to  test  experimentally 
the  exactness  of  it,  owing  to  the  impossibility  of  completely 
preventing  secondary  and  subordinate  reactions  at  the  elec- 
trodes, such  as  the  liberation  of  small  amounts  of  hydrogen 
at  the  catJtjflde  in  the  precipitation  of  metals.  A  careful 
study  of  the  best  conditions  for  eliminating  such  secondary 
actions  has  been  made  in  the  case  of  the  electrolyses  of 
copper  sulphate  and  silver  nitrate  solutions,  however ;  and 
the  ratio  of  the  weights  of  copper  and  silver  precipitated 
under  these  conditions  by  the  same  current  when  passed 
through  the  two  solutions  in  series  has  recently  been  accu- 
rately determined,  and  found  to  be  I  /  3.3940.  The  ratio  of 
the  chemical  equivalents  of  these  two  elements,  as  determined 
by  the  most  exact  chemical  analyses  of  their  compounds,  is 
107.93  /  \  63.60,  or  i  /  3.3940.  The  two  ratios  are  therefore 
identical,  as  Faraday's  Law  requires. 

The  quantity  of  electricity  which  produces  at  each  elec- 
trode a  change  involving  one  chemical  equivalent,  that  is, 
the  numerical  value  of  the  fundamental  constant  £  in  the 
algebraic  expression  of  Faraday's  Law,  is  called  the  electro- 
chemical constant.  It  has  been  determined  by  combining- 
measurements  of  the  quantity  of  silver  precipitated  from  a 
silver  nitrate  solution  with  measurements  of  the  time  and 
current-strength  (made  with  a  galvanometer  or  electro- 
dynamometer)  employed  in  the  precipitation.  Its  most 
probable  value  is  96  600  coulombs. 


120         GENERAL   PRINCIPLES  OF  PHYSICAL  SCIENCE. 

With  the  help  of  this  constant  the  amount  of  electro- 
lytic decomposition  produced  by  the  passage  of  any  definite 
quantity  of  electricity,  and  also  the  quantity  of  electricity 
produced  by  a  definite  voltaic  action,  can  be  calculated. 
For  example,  suppose  it  were  desired  to  determine  how  much 
bismuth  could  be  precipitated  out  of  a  bismuth  chloride 
(BiClg)  solution  by  passing  a  current  of  2.  amperes  through 
it  for  one  hour.  In  this  case,  the  quantity  of  electricity  passed 
(Q=IT)  is  2  X  3600  or  7200  coulombs;  therefore,  the  num- 
ber of  equivalents  precipitated  ( jv  =  Q  /  Q)  is  7200  /  96600  or 
0.0745,  and  the  number  of  grams  precipitated  (m  =  N A)  is 
0.0745  X  69.33  or  5.167,  since  the  equivalent  weight  (A  )  is 
one-third  of  the  combining  weight  208.0  grams,  or  69.33 
grams. 

34.  Heat  Energy.  —  Heat  energy  has  been  defined 
to  be  the  form  of  energy  that  is  manifested  when  bodies 
are  placed  in  communication  with  surroundings  of  different 
temperature.  Temperature  is  therefore  its  intensity-factor, 
as  has  been  already  stated.  Its  capacity-factor  is  called 
heat-capacity  (//) :  it  is  that  which  determines  the  quantity  of 
heat  (dQ)  which  produces  a  definite  increase  of  temperature 
(dT)  in  the  body  or  system  under  consideration ;  that  is, 
H  =  dQ  I  dT.  This  resolution  of  heat-energy  into  factors 
is  an  application  of  the  general  equation  i=-dE  / '  dc  to  a 
case  where  i  and  dE  are  first  independently  determined. 

The  heat-capacity  of  different  homogeneous  bodies  com- 
posed of  the  same  chemical  substance  is  directly  proportional 
to  their  masses  or  weights  ;  that  is,  H  =  Hm,  where  H  is 
the  heat-capacity  of  one  gram  of  the  substance,  which  is 
commonly  called  its  specific  heat,  but  which  is  more  appro- 
priately designated  its  specific  heat-capacity.  Its  value  varies 
with  the  temperature,  but  ordinarily  only  by  a  small  amount. 

Since  a  body  may  be  heated  under  various  conditions, 
and  since  its  heat-capacity  may  vary  correspondingly,  the 
conditions  of  heating  must  be  taken  into  consideration.  The 
two  most  important  cases  are  that  in  which  the  body  expands 


GENERAL  PRINCIPLES  RELATING    TO   ENERGY,         121 

under  a  constant  pressure,  usually  the  pressure  of  the  atmos- 
phere, and  that  in  which  the  volume  of  the  body  is  kept 
constant  by  the  application  of  a  suitable  external  pressure. 
The  corresponding  heat-capacities  are  called  the  heat-capacity 
at  constant  pressure  (Hf  )  and  that  at  constant  volume  (Hv  ). 
In  the  case  of  solid  or  liquid  bodies,  it  is  always  understood, 
when  not  otherwise  specified,  that  the  heating  takes  place 
under  the  constant  pressure  of  the  atmosphere.  In  the  case 
of  gases,  the  conditions  of  heating  must  always  be  stated. 

It  is  found,  however,  in  some  cases  —  for  example,  in 
that  of  a  solid  substance  at  the  temperature  of  its  melting- 
point,  or  of  a  liquid  at  that  of  its  boiling-point  —  that  no 
change  of  temperature  takes  place  when  heat  is  imparted  to 
the  body ;  but  in  such  cases  a  change  in  the  body's  state  of 
aggregation  occurs,  whereby  its  internal  energy  increases ; 
and  the  same  amount  of  heat  is  evolved  when  the  body 
returns  to  its  original  condition.  The  heat  absorbed  in 
such  changes  of  state  taking  place  at  constant  temperature 
is  called  the  heat-effect  of  the  fusion,  vaporization,  or  subli- 
mation (L).  That  absorbed  by  the  change  in  state  of  one 
gram  of  the  substance  is  commonly  called  the  latent  heat,  or 
simply  the  heat,  of  fusion,  of  vaporization,  or  of  sublimation, 
(L),  of  the  substance ;  but,  in  accordance  with  scientific 
usage  in  the  case  of  other  specific  properties,  it  is  more 
rational  to  designate  it  the  specific  heat  of  fusion,  vapor- 
ization, or  sublimation.  It  is  understood,  when  not  specified, 
that  the  change  in  state  takes  place  under  the  atmospheric 
pressure.  An  analogous  phenomenon  is  the  dissolving  of 
one  substance  in  another ;  and  the  heat  absorbed  when  one 
gram  dissolves  is  called  the  (specific]  heat  of  solution  of  the 
substance.  When  not  otherwise  specified,  it  is  to  be  under- 
stood that  the  substance  dissolves,  under  the  usual  conditions, 
in  such  an  amount  of  solvent  that  a  saturated  solution  is 
formed. 

Chemical  changes  attended  by  heat-effects,  as  well  as 
changes  in  physical  state,  may,  of  course,  also  take  place 


122         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

within  a  system.  The  heat  absorbed  by  a  system  as  the 
result  of  a  chemical  change  in  it  is  called  the  heat-effect  of  the 
reaction.  In  order  to  express  this  heat-effect  as  a  specific 
property  of  the  reacting  substances,  it  is  customary  in  ener- 
getic considerations  to  regard  as  the  heat  of  reaction  (q}  that 
absorbed  when  the  number  of  mols  undergo  change  that  are 
represented  by  the  simplest  chemical  equation  that  can  be 
written,  using  the  molecular  formulas  of  the  compounds  in- 
volved and  avoiding  fractional  coefficients.  Thus,  by  the  heat 
of  reaction  or  of  combination  of  hydrogen  and  oxygen  is  meant 
that  corresponding  to  the  equation,  2  H2  +  O2  =  2  H2O,  that 
is,  that  absorbed  by  the  union  of  4.03  grams  of  hydrogen  with 
32.00  grams  of  oxygen. 

In  the  case  where  a  system  undergoes  the  same  change 
in  state  (whether  physical  or  chemical)  at  two  different  tem- 
peratures, and  no  external  work  is  done,  the  First  Law  of 
Energetics  gives  a  simple  relation  between  the  quantities  of 
heat  (QTl  and  (2rz)  absorbed  by  the  change  at  the  two  tem- 
peratures (7i  and  7"2)  and  the  mean  values  between  those 
temperatures  of  the  heat-capacities  (Hz  and  Hn)  of  the  sys- 
tem in  the  two  states  (those  in  which  it  is  before  and  after 
the  change,  respectively).  This  relation  is  derived  by  the 
following  consideration.  Suppose  we  start  with  the  system 
in  the  state  /  at  the  temperature  7i,  and  bring  it  into  the 
state  IT  at  the  temperature  Tz.  This  result  can  be  effected 
in  two  ways,  either  by  causing  the  change  in  state  to  take 
place  at  7i  and  then  changing  the  temperature  of  the  system 
to  TZ,  or  by  changing  the  temperature  first  to  T2  and  then 
causing  the  change  in  state  to  take  place  at  that  temperature. 
The  quantities  of  heat  absorbed  from  the  surroundings  in  the 
two  processes  are, 

QTi  +  Hu  (T,  -  7\)  and  H,  (T,  -  7\)  +  QTf 

respectively.  And  since  the  initial  and  final  states  of  the 
system  are  the  same,  the  initial  and  final  values  of  its  internal 
energy  must  be  the  same,  and  therefore  the  quantities  of  heat 
absorbed  in  the  two  processes  of  converting  it  from  one  of 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         123 

these  states    to  the  other  must    be  equal  ;   for  it   has   been 
assumed  that  no  external  work  is  done.     Hence  it  follows  that  : 


It  will  be  seen  from  this  equation  that  the  heat-effect  attend- 
ing the  change  of  a  system  from  one  state  to  another  is  inde- 
pendent of  the  temperature  when  its  heat-capacities  in  the 
two  states  are  equal,  and  only  in  this  case  ;  and  conversely. 

As  an  example,  consider  the  following  application  of  the 
equation  :  The  heat  evolved  by  the  combination  of  2  grams 
of  hydrogen  with  16  grams  of  oxygen  in  a  closed  vessel  at 
20°  is  67600  cal.,  and  the  specific  heat-capacities  (at  constant 
volume)  of  hydrogen,  oxygen,  and  liquid  water  are  2.41,  0.155, 
and  i.oo,  respectively.  By  substituting  these  values  in  the 
equation,  the  change  of  the  heat-effect  of  the  combination  of 
the  two  gases  with  the  temperature,  and  its  value  (QTt)  at  any 
other  temperature,  such  as  80°,  are  calculated  as  follows  : 
QTi  _  QTt  —  (2  X  2.41  +  16X0.155  —  i8X  i.oo)  (80  —  20) 
=  —  642  cal.  ;  whence  follows,  —  £7^  =  66958  cal. 

The  above  equation  is  of  considerable  importance  from 
an  experimental  standpoint,  since  it  is  not  practicable,  owing 
to  the  errors  from  radiation,  to  make  accurate  calorimetric 
measurements  of  the  heat-effect  attending  physical  changes 
or  chemical  reactions  at  temperatures  far  removed  from  the 
room-temperature,  and  since,  on  the  other  hand,  there  is 
usually  no  difficulty  in  determining  the  heat-capacity  of 
systems  between  the  latter  temperature  and  any  higher  or 
lower  one. 

In  case  quantities  of  work,  WTi  and  WTf  are  pro- 
duced by  the  change  in  state  at  the  two  temperatures  in 
addition  to  the  quantities  of  heat  absorbed  (no  work,  however, 
being  involved  in  the  changes  of  the  temperature  of  the 
system  in  either  state),  the  equation  expressing  the  identity 
of  the  change  in  internal  energy  in  the  two  processes 
obviously  becomes  : 

«2r,  -  »V,)  -  «2r.  -  WT)  =  (H,  ~  «„)  (71  ~  Z"i). 


124          GENERAL  PRINCIPLES   OF  PHYSICAL  SCIENCE. 

If,  in  changing  the  temperature  from  7\  to  7^,  a  change 
in  state  of  aggregation  takes  place  within  the  system,  the 
heat-effect  corresponding  to  this  change,  and  also  the  heat- 
capacities  of  the  system  before  and  after  it,  must  be  taken 
into  consideration  in  calculating  the  change  in  the  heat  of 
reaction  with  the  temperature.  Although  the  special  equa- 
tion derived  above  is  not  applicable  in  this  case,  yet  the  same 
general  principle  consisting  in  equating  the  quantities  of  heat 
absorbed  in  two  processes  of  transforming  the  system  from  a 
definite  initial  state  at  the  one  temperature  to  a  definite  final 
state  at  the  other  temperature  is  to  be  here  employed.  Thus, 
in  the  example  just  considered,  if  the  heat-effect  of  the  com- 
bination of  the  hydrogen  and  oxygen  to  form  water-vaflvr  at 
125°  were  to  be  calculated,  it  would  be  necessary  to  know 
the  specific  heat  of  vaporization  of  water  at  some  temperature, 
say  at  100°,  and  the  specific  heat-capacity  of  its  vapor  between 
that  temperature  and  125°.  Since  the  former  value,  when  no 
external  work  is  done,  is  498  cals.,  and  the  latter  value  is 
0.30,  the  heat  of  combination  at  125°  is  —  57800  cal.,  as  will 
be  seen  on  making  the  calculation. 

35.  Chemical  Energy.  —  Chemical  energy  has  been 
defined  to  be  the  energy  that  systems  possess  in  virtue  of 
the  tendency  of  the  substances  composing  them  to  undergo 
transformations  into  other  substances.  The  total  decrease 
in  the  internal  energy  of  a  system  in  which  a  chemical 
change  takes-  place  is  readily  determined  by  measuring  the 
heat-effect  and  work  produced  in  the  surroundings ;  but 
this  cannot  be  placed  equal  to  the  decrease  in  chemical 
energy  :  for  a  chemical  change  is  always  accompanied,  not 
only  by  a  change  in  the  chemical  energy,  that  is,  in  the 
energy  which  gives  rise  to  the  tendency  to  the  change, 
but  also  by  changes  in  other  forms  of  energy,  namely,  in 
the  cohesion  and  disgregation  energies  of  the  system,  or, 
otherwise  expressed,  in  its  heat  energy,  as  is  shown  by  the 
change  in  the  system's  heat-capacity  which  commonly 
accompanies  the  chemical  change.  In  certain  special 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         125 

cases,  to  be  sure,  the  chemical  energy  can  be  separately 
determined  ;  namely,  where  electrical  energy  is  produced 
by  the  chemical  change  and  the  change  can  be  made  to 
take  place  in  the  reverse  direction  by  imparting  an  equal 
amount  of  electrical  energy  to  the  system. 

The  resolution  of  chemical  energy  into  factors  has  little 
significance  from  an  experimental  standpoint,  since  in  most 
cases  the  numerical  value  of  the  energy  itself  is  unknown ; 
but  it  is  sometimes  convenient  in  theoretical  discussions. 
The  definitions  of  the  factors  that  have  been  proposed  do 
not,  however,  need  to  be  presented  here. 

36.  Radiant  Energy.  —  Radiant  energy  has  been  de- 
fined to  be  the  form  in  which  energy  is  transmitted  from 
one  body  to  another  without  the  mediation  of  matter,  and  it 
has  been  stated  that  there  are  different  varieties  of  it  called 
light,  radiant  heat,  and  electromagnetic  radiations.  These 
have,  however,  so  many  common  characteristics  that  they 
are  considered  to  be  of  essentially  the  same  nature.  For 
instance,  it  has  been  proved  that  the  velocities  with  which 
light  and  electromagnetic  radiations  are  transmitted  through 
space  are  identical  within  the  limits  of  the  experimental 
errors,  the  value  being  3.00  X  io10  centimeters,  or  300000 
kilometers  per  second.  Moreover,  these  three  classes  of 
radiations  undergo  reflection  and  refraction  in  accordance 
with  the  same  laws,  and  they  all  exhibit  the  phenomena  of 
interference  and  polarization. 

The  phenomenon  of  interference,  that  is,  the  fact  that 
two  rays  when  superposed  under  suitable  conditions  anni- 
hilate each  other  (two  rays  of  light,  for  example,  producing 
darkness),  has  led  to  the  conclusion  that  radiant  energy  is 
transmitted  in  the  form  of  waves ;  and,  to  facilitate  the 
comprehension  of  the  phenomena  by  making  analogies  pos- 
sible with  the  wave  phenomena  exhibited  by  matter,  it  is 
conceived  that  the  waves  are  propagated  in  a  hypothetical 
medium,  called  the  ether,  and  are  the  result  of  disturbances 
of  some  kind  or  other  produced  in  it.  The  length  of  the 


126         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

waves  of  radiant  energy,  or  more  briefly  the  wave-length  (A,) 
of  the  rays,  can  be  derived  from  the  measurement  of  inter- 
ference effects ;  and  it  is  found  in  this  way  that  its  values 
are  widely  different  in  the  cases  of  light  or  heat  rays  and  of 
electromagnetic  rays,  and  that  there  are  smaller,  though 
considerable  differences  in  wave-length  in  the  case  of  differ- 
ent rays  of  the  same  class,  as  in  that  of  light  rays  of 
different  colors.  Thus,  the  smallest  wave-length  that  has 
been  measured,  that  of  a  ray  thrown  far  beyond  the  violet 
end  of  the  visible  spectrum  and  detected  by  photographic 
means,  is  about  o.io  p  (p  representing  one  micron  which 
is  equal  to  io6  meters),  the  wave-lengths  of  the  rays  at  the 
two  limits  of  the  visible  spectrum  are  0.36  and  0.81  yu,  and 
the  largest  wave-length  measured  for  any  heat  ray  is  50  ^, 
while  electromagnetic  rays  have  been  produced  whose  wave- 
lengths vary  from  0.4  cm.  to  many  kilometers.  The  similarity 
in  kind  of  almost  all  the  effects  produced  by  these  varieties 
of  radiant  energy  has  led  to  the  conclusion  that  they  differ 
only  in  wave-length,  and  therefore  that  the  specification  of 
this  suffices  to  characterize  them  qualitatively. 

The  quantity  of  the  effects  produced  by  radiant  energy 
is,  however,  determined  by  another  characteristic  of  it, 
which  is  termed  its  intensity  (i),  and  which  is  defined,  in 
general,  as  the  quantity  of  radiant  energy  passing  in  a 
second  through  a  square  centimeter  of  area  perpendicular 
to  the  rays.  This  may  be  determined  in  the  case  of  light 
and  heat  rays  by  receiving  them  during  a  time  T  upon  a 
blackened  surface  of  area  s  perpendicular  to  the  rays,  by 
which  they  are  absorbed  and  converted  into  heat,  and 
measuring  the  quantity  of  heat  Q  produced ;  for  by  defini- 
tion, i  =  Q  I  ST.  For  measuring  the  intensity  of  white 
light,  as  in  photometry,  a  conventional  system  is  usually 
employed,  the  illumination  produced  by  the  light  upon  a 
screen  being  made  equal  to  that  produced  by  some  standard 
source  of  light  by  varying  its  distance  from  the  screen ; 
since  the  intensity  of  light  radiated  uniformly  in  all  direc- 


GENERAL   PRINCIPLES  RELATING    TO  ENERGY.         127 

tions  varies  inversely  as  the  square  of  the  distance  of  the  re- 
ceiving surface  from  the  source,  the  unknown  intensity  can 
be  readily  calculated  and  expressed  in  terms  of  a  conven- 
tional unit. 

The  only  phenomena  connected  with  radiant  energy 
that  require  consideration  here  are  those  of  the  refraction, 
polarization,  absorption,  and  emission  of  light ;  for  these 
alone  have  important  relations  to  the  chemical  nature  of 
substances. 

When  a  ray  of  light  passes  through  a  surface  separat- 
ing one  medium  from  another,  its  direction  is  changed  in 
accordance  with  the  following  law,  known  as  SnelVs  Law  of 
Refraction :  the  incident  and  refracted  rays  and  the  normal 
to  the  surface  at  the  point  of  incidence  are  in  the  same  plane, 
and  the  sines  of  the  angles  which  the  two  rays  make  with 
that  normal  bear  a  constant  ratio  to  each  other.  That  is,  if 
the  incident  and  refracted  rays  make  with  the  normal, 
angles  of  0f  and  6  r,  respectively,  sin  Q{  /  sin  6r  =  v,  where 
v  is  a  quantity,  called  the  index  af  refraction,  constant  with 
reference  to  variations  of  0f  and  0r)  but  variable  with  the 
character  of  the  two  media,  especially  with  their  densities 
and  chemical  nature,  and  with  the  wave-length  of  the  light. 
When  the  incident  ray  passes  from  a  vacuum  into  a  second 
medium,  the  corresponding  value  of  v  is  called  the  absolute 
index  of  refraction  of  the  medium.  Ordinarily,  however, 
the  index  of  refraction  of  a  medium  is  determined  and 
stated  with  reference  to  the  entrance  of  the  ray  into  it 
from  air ;  yet,  since  the  absolute  index  for  air  under  normal 
conditions  is  1.0003,  tne  absolute  index  of  a  medium  is  only 
0.03  per  cent,  greater  than  its  index  with  reference  to  air, 
and  may  usually  be  considered  identical  with  it.  It  has 
been  proved  experimentally,  and  has  also  been  deduced 
from  the  Wave  Theory  of  Light,  that  the  index  of  refrac- 
tion of  a  medium  is  equal  to  the  ratio  of  the  velocity  ui 
of  light  in  air  to  its  velocity  u2  in  the  medium  ;  that  is, 
v  =  vl/  v2. 


128         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

When  a  ray  of  ordinary  light  passes  perpendicularly 
through  a  plate  of  tourmaline  properly  cut  with  reference 
to  the  axes  of  the  crystal,  or  is  reflected  from  a  mirror  of 
transparent  material  (such  as  glass)  at  a  suitable  angle,  it 
is  found  to  have  acquired  the  new  characteristic  of  produc- 
ing different  effects  upon  its  different  sides  ;  for,  when  the 
same  ray  is  passed  perpendicularly  through  a  second  tour- 
maline plate,  or  is  reflected  from  a  second  similar  mirror, 
its  final  intensity  is  found  to  vary  when  the  plate  is  rotated 
in  a  plane  at  right  angles  to  the  ray,  or  when  the  mirror  is 
turned  around  the  ray  keeping  the  angle  of  incidence  con- 
stant, and  to  have  a  minimum  value  of  zero  and  a  maximum 
value  at  two  positions  of  the  plate  or  mirror  that  are  at 
right  angles  to  each  other.  Light  which  produces  such 
different  effects  in  the  various  directions  at  right  angles  to 
the  ray  and  a  zero  effect  in  one  of  those  directions  is  said 
to  be  plane-polarized,  or  simply  polarized.  In  order  to 
facilitate  the  expression  of  these  directions,  it  is  agreed  to 
select  as  a  basis  of  reference  the  plane  which  embraces  the 
incident  ray  and  the  normal  to  the  mirror  when  it  is  in 
the  position  where  the  maximum  amount  of  light  is  reflected, 
and  to  designate  this  the  plane  of  polarization. 

When  a  ray  of  polarized  light  passes  through  bodies 
of  a  certain  crystalline  structure,  such  as  some  varieties  of 
quartz,  or  through  bodies  consisting  of  certain  chemical 
substances,  such  as  a  tube  of  oil  of  turpentine  or  a  solution 
of  cane-sugar,  its  plane  of  polarization  is  rotated  through  a 
greater  or  less  angle  in  the  one  direction  or  the  other. 
This  may  be  determined  by  examining  the  emerging  ray 
with  a  glass  mirror,  a  tourmaline,  or  other  suitable  analyzer. 
Substances  which  rotate  the  plane  of  polarization  are  called 
optically  active  substances :  those  that  rotate  it  to  the  right 
when  looking  in  the  direction  in  which  the  light  is  traveling 
are  called  right-rotating  or  dextro-rotatory  substances  ;  those 
which  produce  rotation  in  the  opposite  direction,  left-rotating 
or  leavo-rotatory  substances.  The  angle  of  rotation  ex  is 


GENERAL  PRINCIPLES  RELATING   TO  ENERGY.         129 

found  in  the  case  of  a  homogeneous  body  to  be  exactly  pro- 
portional to  the  length  /  of  the  layer  traversed  by  the  ray, 
and  to  increase,  though  not  always  proportionally,  with  the 
density  or  concentration  c  of  the  active  substance ;  that  is, 
oc  =  «  /<:,  where  a  is  a  quantity,  called  the  specific  rotatory 
power,  defined  by  this  equation  itself,  constant  with  refer- 
ence to  variations  of  /,  and  usually,  though  not  always, 
approximately  so  with  reference  to  those  of  cy  and  variable 
in  a  high  degree  with  the  chemical  substance  and  in  a 
smaller  degree  with  the  temperature  and  other  physical 
conditions,  and  with  the  wave-length  of  the  light.  In  ex- 
pressing numerical  values  of  the  specific  rotatory  power, 
it  is  customary  to  employ  the  decimeter  as  the  unit  of 
length  and  the  gram  per  cubic  centimeter  as  the  unit 
of  concentration. 

While  only  a  very  small  proportion  of  known  sub- 
stances have  in  themselves  the  power  of  rotating  the  plane 
of  polarized  light,  all  transparent  substances  acquire  it  in 
variable  degree  when  placed  in  a  magnetic  field,  the  effect 
being  strongest  when  the  lines  of  force  in  the  field  are 
parallel  to  the  ray  of  light,  and  zero  when  they  are  per- 
pendicular to  it.  For  instance,  when  a  ray  of  plane-polar- 
ized light  is  passed  through  a  tube  of  water  around  which 
there  is  a  helix  of  wire  through  which  an  electric  current 
is  flowing,  the  plane  of  polarization  of  the  emerging  ray  is 
found  to  be  different  from  that  of  the  entering  ray.  This 
phenomenon  is  known  as  magnetic  rotation  of  the  plane  of 
polarized  light.  The  angle  of  rotation  is  proportional  to  the 
strength  of  the  current,  or  to  that  of  the  magnetic  field 
which  it  produces,  and  to  the  length  of  the  layer  submitted 
to  its  influence  and  traversed  by  the  ray.  The  direction  of 
the  rotation  is  reversed  when  either  that  of  the  electric 
current  or  that  of  the  ray  is  reversed. 

When  a  beam  of  light  of  any  definite  wave-length  is 
passed  through  a  body,  in  general  the  intensity  of  the 
emerging  beam  is  found  to  be  less  than  that  of  the  enter- 


130         GENERAL  PRINCIPLES  OF  PHYSICAL   SCIENCE. 

ing  beam,  owing  to  the  absorption  of  a  portion  of  the  light, 
and  the  conversion  of  it  into  heat,  within  the  body.  It  is 
found  that  a  constant  fraction  of  the  light  that  enters  each 
successive  section  of  a  homogeneous  body  is  absorbed  in  that 
section,  all  the  sections  being  considered  to  be  of  equal  and 
indefinitely  small  thickness ;  or,  expressed  in  mathematical 
form,  if  i  is  the  intensity  of  the  beam  at  any  distance  /  from 
the  point  of  entrance,  and  —  di  is  its  decrease  in  intensity 
on  traversing  the  further  distance  dl,  then,  — di  I  i  =  f3dlt 
where  j3  is  a  quantity  termed  the  coefficient  of  absorption, 
constant  with  reference  to  variations  of  i  and  /,  but  variable 
with  the  nature  and  physical  state  of  the  substance  and  with 
the  wave-length  of  the  light.  On  integration,  this  equation 
gives,  log  (^  /  4a)  =  £  /,  where  ^  and  iz  are  the  intensities  of 
the  entering  and  emerging  beams,  respectively,  and  /  is  the 
total  length  of  the  layer  traversed. 

It  is  found  that  the  absorption-coefficient  of  gaseous 
substances  at  moderate  pressure  is  very  large  for  certain 
special  wave-lengths,  which  vary  with  the  nature  of  the  gas, 
and  practically  zero  for  all  other  wave-lengths,  so  that  if  a 
beam  of  white  light,  such  as  is  emitted  by  an  incandescent 
solid,  be  passed  through  such  gases  and  be  resolved  by  a 
spectroscope,  its  spectrum  is  found  to  be  crossed  by  a  num- 
ber of  sharply  defined  dark  lines  corresponding  to  the  wave- 
lengths for  which  practically  complete  absorption  takes 
place.  The  values  of  these  wave-lengths  have  been  far 
more  extensively  determined  and  are  of  greater  scientific 
importance  than  the  values  of  the  absorption-coefficients 
for  definite  wave-lengths.  When  the  density  of  gases  is 
increased  by  compression,  the  lines  of  their  absorption- 
spectra  become  broader  and  less  sharply  defined  at  the 
edges.  And  in  some  liquids  and  solids  the  absorption 
takes  place  in  such  a  manner  as  to  produce  in  the  spectrum 
one  or  more  broad  bands,  each  possessing  a  somewhat 
indefinite  maximum  of  darkness  and  ill-defined  limits. 
In  other  liquids  the  absorption-coefficient  increases  or 


GENERAL   PRINCIPLES  RELATING    TO  ENERGY.         131 

decreases  continuously  with  increasing  wave-length,  or  de- 
creases to  a  minimum  at  some  value  of  the  wave-length 
and  then  increases  continuously. 

The  phenomena  of  the  absorption  of  light  have  addi- 
tional interest,  inasmuch  as  they  are  closely  connected  with 
the  color  of  bodies,  which  is  a  property  that  arises  usually 
from  the  unequal  absorption  out  of  white  light  of  the  rays 
of  different  wave-lengths,  the  impression  produced  on  the  eye 
being  that  caused  by  the  unabsorbed  rays.  That  selective 
absorption  is  the  cause  of  the  color  of  objects  that  are  seen 
by  transmitted  light  is  obvious  :  but  this  is  known  to  be,  in 
most  cases,  also  the  source  of  the  color  of  objects  viewed  by 
reflected  light ;  for  the  light,  before  it  is  reflected  to  the 
eye,  generally  traverses  a  layer  of  the  substance  of  greater 
or  less  thickness.  Thus,  the  colors  of  a  piece  of  cobalt 
glass  and  of  a  potassium  dichromate  solution  on  the  one 
hand,  and  of  pigments  on  the  other,  are  alike  due  to  se- 
lective absorption.  In  some  cases,  to  be  sure,  as  in  those 
of  metals  and  solutions  of  many  organic  dyestuffs,  the  color 
is  due  to  unequal  reflection  of  the  rays  of  different  wave- 
lengths at  the  surface  of  the  body,  and  is  then  termed 
surface-color.  Bodies  possessing  surface-color  must  obvi- 
ously show  different  colors  according  as  they  are  viewed  by 
reflected  or  transmitted  light ;  this  is  the  case,  for  example, 
with  gold,  which  is  yellow  by  reflected  light,  and  green,  in 
very  thin  layers,  by  transmitted  light,  and  with  a  strong 
solution  of  fuchsine,  which  is  green  by  reflected,  and  red  by 
transmitted  light. 

The  emission  of  light  by  heated  bodies  is  also  closely 
connected  with  their  powers  of  absorption  and  surface- 
reflection  ;  for,  according  to  Kirchhoff's  Law,  the  rays  of 
various  wave-lengths  are  emitted  by  a  heated  body  in  the 
same  proportion  in  which  they  are  absorbed  by  it  at  the  same 
temperattire  when  they  fall  upon  it  from  an  external  source. 
In  this  statement,  the  proportion  absorbed  signifies  the 
proportion  of  the  incident  ray  that  is  neither  reflected  nor 


132         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

transmitted.  For  example,  a  heated  mass  of  gas  emits 
most  copiously  those  rays  that  are  absorbed  most  com- 
pletely when  white  light  from  an  external  source  is  passed 
through  it,  and  does  not  emit  at  all  those  rays  that  are 
transmitted  through  it  without  absorption ;  consequently, 
its  emission-spectrum  consists  of  bright  lines  corresponding 
exactly  in  position  to  the  dark  lines  in  its  absorption- 
spectrum.  A  piece  of  ruby  glass,  which  appears  red  by 
transmitted  light  owing  to  absorption  of  the  green  rays, 
emits  green  light  when  heated  to  incandescence  ;  and  a 
mass  of  copper,  which  appears  red  owing  to  surface-reflec- 
tion of  the  red  rays,  also  emits  green  light  when  heated. 

37.  The  Internal  Energy  of  Gases.  Experiments  of 
Gay-Lussac  and  of  Joule  and  Thomson.  —  In  addition  to 
the  principles  already  discussed  relating  to  the  volume  energy 
of  gases  and  its  factors,  an  extremely  important  law  in  regard 
to  their  internal  energy  has  been  discovered.  This  may  be 
stated  as  follows  :  the  internal  energy  of  a  definite  quantity  of 
any  gas  at  a  definite  temperature  is  independent  of  its  pres- 
sure or  volume,  provided  the  pressure  is  small. 

The  validity  of  this  law  was  first  established  by  a  series 
of  experiments  by  Gay-Lussac,  in  which  a  gas,  at  a  pressure  of 
one  atmosphere  or  half  an  atmosphere,  contained  in  a  large 
glass  balloon  was  caused  to  expand  suddenly,  by  opening  a 
cock,  into  another  balloon  of  the  same  volume  in  which  a 
nearly  perfect  vacuum  had  been  produced.  The  temperatures 
within  the  two  balloons,  which  were  made  equal  before  an  ex- 
periment was  performed,  were  carefully  determined  by  means 
of  delicate  thermometers  after  the  expansion  had  taken  place. 
It  was  found  that  the  rise  of  temperature  which  took  place  in 
the  balloon  originally  empty  was  always  equal,  within  the 
experimental  error,  to  the  fall  of  temperature  occurring  in  the 
other  balloon,  so  that,  if  the  gas  in  the  two  balloons  had  been 
mixed,  the  original  temperature  would  have  been  reproduced 
without  any  energy  being  imparted  to  or  taken  up  from  the 
surroundings,  thus  proving  that  the  energy  within  a  gas  is 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         133 

unaffected  by  a  change  of  volume  taking  place  at  constant 
temperature. 

Further  experiments  confirming  the  law  were  made  by 
Joule,  who  substituted  two  copper  cylinders  for  the  glass 
balloons,  immersed  the  whole  apparatus  in  water,  and  deter- 
mined the  temperature  of  the  latter  before  and  after  the 
expansion  of  air  contained  in  one  cylinder  at  22  atmospheres' 
pressure  into  the  other  empty  cylinder.  No  change  of  tem- 
perature could  be  detected. 

This  law,  taken  in  connection  with  the  Law  of  the 
Conservation  of  Energy,  leads  to  two  other  important  laws 
in  regard  to  the  energy-relations  of  gases  under  small 
pressures. 

The  first  of  these  is  as  -follows  :  the  heat  absorbed  from 
the  surroundings  by  a  gas  kept  at  constant  temperature  while 
expanding  against  an  external  pressure  is  equivalent  to  the 
work  done  by  it.  Since  the  energy  of  the  gas  itself  remains 
unchanged,  this  law  is  an  obvious  consequence  of  the  First 
Law  of  Energetics.  That  is,  since  the  increase  of  energy- 
of  the  system  under  consideration  (the  gas)  is  equal  to  zero, 
the  equation  expressing  the  First  Law,  U*  —  £/i  =  Q  —  Wr 
becomes  Q  =  W.  A  gas  under  small  pressure  is  therefore  a 
system  by  means  of  which  heat  can  be  quantitatively  trans- 
formed into  work  at  constant  temperature. 

The  second  law  deducible  from  the  fundamental  one  first 
considered  is  that  the  heat-capacity  of  a  definite  quantity  of  a 
gas  is  independent  of  its  pressure  or  volume.  This  follows 
at  once  from  the  principle  stated  in  §  34  in  regard  to  the  rela- 
tion between  the  heat-capacities  of  a  system  in  two  different 
states,  and  the  heat-effects  that  accompany  the  change  from 
one  state  to  the  other  at  different  temperatures ;  for,  since  the 
expansion  of  a  gas  when  no  external  work  is  done  is  accom- 
panied by  the  same  heat-effect  at  any  two  different  tempera- 
tures, namely,  by  no  heat-effect  at  either  temperature,  the 
heat-capacities  of  the  compressed  and  expanded  gas  are  equal. 

Later  experiments  of  Joule  and  Thomson,  made  by  a, 
more  exact  method,  show,  however,  that  these  laws  are  sub- 


134  GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

ject  to  deviations  of  the  same  order  of  magnitude  as  those 
which  affect  Boyle's  Law.  In  these  experiments,  which  are 
commonly  known  as  the  porous-plug  experiments,  the  gas  to 
be  investigated  was  driven  by  means  of  a  force-pump,  at  a 
perfectly  uniform  rate,  first  through  a  coil  of  pipes  surrounded 
with  water,  by  which  it  was  brought  to  a  constant  tempera- 
ture, and  then  through  a  boxwood  tube  containing  a  porous 
plug  of  unspun  silk  or  cotton-wool  and  covered  with  insulating 
material,  which  prevented  any  flow  of  heat  to  or  from  the 
surroundings.  After  issuing  from  the  plug,  the  gas  was 
allowed  to  escape  into  the  atmosphere,  or  collected  in  a  gas- 
ometer at  atmospheric  pressure.  The  pressure  of  the  gas 
before  it  entered  the  plug  was  measured  with  a  manometer ; 
this  pressure  could  be  varied  in  separate  experiments  by  using 
a  more  or  less  compact  plug  and  by  varying  the  quantity  of 
gas  forced  through  it  per  second.  After  the  gas  had  been 
flowing  so  long  that  it  was  in  thermal  equilibrium  with  the 
plug  and  surrounding  tubes,  its  temperature  was  determined 
by  means  of  delicate  thermometers  just  before  it  entered  the 
plug,  and  just  after  it  issued  from  it.  Since  under  these  cir- 
cumstances the  gas  in  expanding  absorbs  no  heat  from  the 
surroundings,  and  does  only  an  inconsiderable  amount  of  work, 
as  will  be  seen  below,  the  internal  energy  of  the '  gas  is 
nearly  the  same  after  its  expansion  as  it  was  before,  and  from 
the  observed  difference  in  its  temperature  and  its  known  heat- 
capacity,  the  change  in  internal  energy  attending  a  corre- 
sponding expansion  of  the  gas  at  constant  temperature  can 
be  accurately  deduced.  Since  the  results  are  of  fundamental 
importance,  the  principles  involved  will  be  here  presented. 

Let  the  temperature,  pressure,  volume,  and  internal  energy 
of  m  grams  of  the  gas  just  before  it  enters  the  porous  plug 
be  *!,  /!,  vlf  and  £/i,  respectively,  and  the  corresponding  mag- 
nitudes after  the  gas  has  issued  from  the  plug  in  the  expanded 
condition  be  /2,  pi,  v*,  and  [72f,  respectively.  Let  the  m 
grams  of  gas  then  be  heated  at  the  constant  pressure /2  from 
/2  to  /!,  whereby  its  volume  becomes  v2,  and  its  internal  energy 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         135 

£/2-  It  is  desired  now  to  determine  the  value  [7Z  —  [7l}  which 
represents  the  increase  in  the  internal  energy  of  m  grams  of 
the  gas  when  at  the  constant  temperature  t\  its  volume  in- 
creases from  ^  to  z/a,  or  its  pressure  decreases  from  p±  to  /2. 
In  order  to  do  this,  we  apply  the  general  expression  of  the 
First  Law,  U*  —  Ul  =  Q  —  W,  successively  to  the  two  pro- 
cesses to  which  the  gas  has  been  subjected.  In  the  first 
process,  Q  =  o  ;  and  W—fav2f  —  p\v\,  as  will  be  evident 
from  a  reference  to  the  accompanying  figure,  and  from  the 
consideration  that  when  the  gas  enters  the  plug  the  volume 
v-i  is  caused  to  disappear  under  the  constant  pressure  /x,  and 
that  when  it  issues  from  the  plug  the  volume  vj  is  produced 


FIG.  5. 
against  the  constant  pressure  /2.     It  follows  therefore  that  : 


In  the  second  process,  in  which  the  gas  is  heated  under 
the  constant  pressure  /3  from  t*  to  tlt  it  is  evident  that 
Q  =  m-ffj(fi—tt),  if  Hp  is  the  specific  heat-capacity  of  the 
gas  at  constant  pressure,  and  that  W=fa  (v*  —  vj)  ;  therefore, 

U*  —Uj  =  m  Hp  (/!  —  /,)  —  /2  (vt  —  vj), 
whence  follows  by  combination  with  the  foregoing  equation, 


The  evaluation  of  this  change  in  internal  energy  obvi- 
ously requires  not  only  a  knowledge  of  the  temperature- 
change  occurring  in  the  porous-plug  experiments,  but  also 
that  of  the  heat-capacity  of  the  gas  at  constant  pressure  and 
of  the  values  of  its  /^-product  at  the  two  pressures  in  ques- 
tion; both  of  these  latter  quantities  have,  however,  been 
independently  determined  for  most  gases  by  direct  measure- 
ments, It  will  be  noted  that  the  term  (p&i  —  pzvz)  would 


136          GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

have  the  value  zero  for  a  gas  that  followed  Boyle's  Law,  and 
that,  according  to  the  statements  of  §  21,  since  here  pl 
is  greater  than  /2,  it  actually  does  have  a  small  positive 
value  for  hydrogen  and  a  small  negative  value  for  all  other 
gases.  The  porous-plug  experiments  of  Joule  and  Thomson 
show,  however,  that  air,  oxygen,  nitrogen,  and  carbon  dioxide 
undergo  on  expansion  a/#//  of  temperature,  which  corresponds 
to  a  positive  value  of  (A  —  /2),  and  that  its  magnitude  is  con- 
siderably more  than  sufficient  to  compensate  the  negative 
value  of  (/i^i J—  P*v<?).  Therefore,  the  internal  energy  of  these 
gases  increases  when  their  volume  increases  and  their  tem- 
perature is  kept  constant.  Incidentally,  it  is  of  interest  to 
note  the  general  magnitude  of  the  fall  of  temperature  ob- 
served;  at  I7°-20°,  this  was  0.26°  with  air,  and  1.15°  with 
carbon  dioxide,  for  a  decrease  of  pressure  of  one  atmosphere. 
With  hydrogen,  on  the  other  hand,  a  slight  rise  of  tempera- 
ture (0.033°  f°r  a  pressure-decrease  of  one  atmosphere)  was 
•observed  in  the  porous-plug  experiments ;  but  its  magnitude 
was  not  quite  sufficient  to  compensate  the  positive  value  of 
(P\v\ — A^)>  so  that,  apparently  for  this  gas  also,  (U* —  £/i) 
has  a  positive  value :  it  should  be  added,  however,  that  in  this 
case  the  calculated  effect  does  not  exceed  the  possible  experi- 
mental error.  ^ 

The  best  idea  of  the  magnitude  of  this  energy-increase 
can  be  obtained  by  comparing  its  value  with  that  of  the 
external  work  done  when  the  gas  expands  against  a  pressure 
sensibly  equal  to  its  own.  Computations  have  shown  that, 
when  the  gas  has  a  pressure  of  one  atmosphere  and  under- 
goes an  infinitesimal  increase  in  volume,  the  ratio  (dU  /  d  W) 
of  the  increase  in  internal  energy  to  the  work  done  has  at 
the  room-temperature  the  value  ^TVtf  in  the  case  of  hydro- 
gen, -^fa  in  the  case  of  air,  and  -fa  in  the  case  of  carbon 
dioxide ;  and  that  it  has  at  92°  the  value  gi^  in  the  case  of 
the  last-named  gas.  The  deviation  from  the  fundamental 
law  stating  the  constancy  of  the  internal  energy  is  therefore 
almost,  if  not  quite,  within  the  experimental  error  in  the  case 


GENERAL   PRINCIPLES  RELATING   TO  ENERGY.          137 

of  hydrogen ;  very  small,  though  undoubtedly  real,  in  the 
case  of  air ;  much  greater  in  the  case  of  carbon  dioxide ; 
and,  in  its  case,  greater  at  the  lower  than  the  higher  temper- 
ature. Thus,  it  is  seen  that  the  deviation  is  greater,  the 
greater  the  deviation  from  Boyle's  Law;  and  this  justifies 
the  inference  that  the  former  deviation  would  be  zero  in  the 
case  of  a  perfect  gas.  In  the  case  of  gases  under  high 
pressure,  the  law  does  not  hold  true  even  as  an  approxima- 
tion; the  cooling  effect  on  expansion,  even  when  no  work 
is  done,  being  then  so  great  that  it  has  been  employed  tech- 
nically in  the  production  of  liquid  air. 

38.  The  Second  Law  of  Energetics.  —  Besides  the 
Law  of  its  Conservation,  another  general  principle  relating  to 
the  transformation  of  energy  has  been  empirically  established. 
It  has  been  found  that,  while  other  forms  of  energy  are 
readily  and  completely  transformed  into  heat,  the  transforma- 
tion of  heat  into  those  other  forms  is  subject  to  certain  limi- 
tations. Thus,  failure  has  attended  all  attempts  to  devise  a 
machine  or  arrangement  of  matter  which  will  do  work  in 
unlimited  amount  merely  by  withdrawing  heat  from  the  sur- 
roundings. An  ideal  process,  in  which  heat  is  considered  to 
be  taken  from  surroundings  of  constant  temperature  and 
transformed  into  work  by  a  system  which  itself  undergoes 
no  permanent  modification,  is  called  perpetual  motion  of  the 
second  kind ;  and  a  most  extensive  experience  has  shown 
that  perpetual  motion  of  the  second  kind  is  impossible. 
Experience  with  processes  taking  place  at  different  temper- 
atures has  led  to  the  conclusion  that  this  principle  is  a  con- 
sequence of  a  still  more  general  natural  law  which  may  be 
expressed  as  follows :  a  process  whose  final  result  is  only  a 
transformation  of  a  quantity  of  heat  into  work  is  an  impos- 
sibility. This  law  is  called  the  Second  Law  of  Energetics ; 
and  this  general  statement  of  it  will  serve  as  a  postulate 
forming  the  basis  of  the  following  considerations  in  regard 
to  it. 

As  a  concrete  illustration  of  the  Law,  it  may  be  men- 


138          GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

tioned  that  a  process  resulting  solely  in  raising  a  weight  and 
cooling  a  reservoir  is  an  impossibility.  Such  a  process  would 
evidently  not  be  a  contradiction  of  the  First  Law ;  for  this 
would  only  require  that  the  work  done  and  heat  absorbed 
be  equal.  It  may  be  further  mentioned,  by  way  of  illustra- 
tion, that,  if  the  Second  Law  were  not  true,  mechanical 
power  could  be  obtained  in  unlimited  quantities  from  the 
vast  amount  of  heat  present  in  the  ocean,  thus  making  un- 
necessary the  fuel  used  in  steamships  as  a  source  of  energy. 

The  law,  of  course,  does  not  imply  that  heat  can  not 
be  transformed  into  work,  but  only  that  its  transformation 
must  be  attended  by  a  change  in  the  condition  of  some  other 
quantity  of  energy.  This  attendant  change  may  be  either 
a  change  in  the  condition  of  the  energy  of  the  surroundings, 
or  in  that  of  the  system  employed  for  the  transformation ; 
but  it  is  always  of  such  a  character  that  the  power  of  pro- 
ducing work  which  the  surroundings  or  the  system  possesses 
is  diminished. 

The  Second  Law  is  based  on  an  experience  so  extensive 
and  varied  that  the  probability  of  meeting  with  an  exception 
to  it  is  extremely  slight ;  and  especially  so,  in  connection 
with  mechanical  or  electrical  processes.  It  is  proved  directly 
by  the  failure  of  all  attempts  to  produce  perpetual  motion 
of  the  second  kind,  and  indirectly  by  the  quantitative  cor- 
respondence of  many  conclusions  drawn  from  it  in  regard  to 
relations  between  different  properties  of  substances,  with  those 
actually  found  to  exist. 

The  Second  Law  is  quite  distinct  in  significance  from  the 
First  Law,  as  may  be  made  clearer  by  contrasting  them. 
The  First  Law  states  that  work  can  not  be  done  by  a  system 
that  undergoes  no  permanent  modification,  except  by  with- 
drawing energy  of  some  form  from  the  surroundings  ;  in  other 
words,  that  perpetual  motion  of  the  first  kind  is  impossible. 
The  Second  Law  states  that  work  can  not  be  done  by  such  a 
system  'merely  by  withdrawing  heat  from  the  surroundings, 
that  is,  that  perpetual  motion  of  the  second  kind  is  impossible. 


GENERAL  PRINCIPLES  RELATING   TO  ENERGY.          139 

The  First  Law  asserts  that  when  a  new  quantity  of  energy 
appears  in  one  form  or  at  one  place,  an  equivalent  quantity  of 
it  of  another  form  or  at  another  place  must  disappear.  The 
Second  Law  asserts  that  when  a  new  quantity  of  work  is 
produced  by  the  transformation  of  heat,  there  must  be  an 
accompanying  change  in  the  condition  of  some  other  quantity 
of  energy  of  such  a  character  that  the  work  it  is  capable  of 
producing  is  correspondingly  diminished. 

39.  Application  of  the  Second  Law  to  Changes 
Taking  Place  at  a  Constant  Temperature.  —  The  change 
attending  the  transformation  of  heat  into  work  may  be  either 
a  change  in  the  condition  of  the  transforming  system  or  in 
that  of  the  energy  in  the  surroundings,  as  was  stated  above. 

The  first  of  these  two  cases  is  well  illustrated  by  the 
expansion  of  a  gas  against  an  external  pressure  sensibly 
equal  to  its  own ;  for  a  definite  quantity  of  heat  is  thereby 
transformed  into  an  equivalent  amount  of  work,  and  the 
expanding  gas,  which  is  the  system  employed  for  the  trans- 
formation, though  it  retains  the  same  quantity  of  energy,  so 
changes  its  condition  (decreases  its  pressure  and  increases  its 
volume)  that  its  power  of  transforming  heat  is  diminished. 

The  second  case  is  that  in  which  the  system  does  not 
undergo  any  permanent  change,  but  is  in  the  same  condition 
at  the  enti  of  the  processes  to  which  it  has  been  subjected  as 
it  was  at  the  beginning.  Such  a  series  of  processes  is  called 
a  cyclical  process.  (It  is  to  be  noted  that  the  term  process  is 
used  both  to  express  the  way  in  which  a  change  in  the  state 
of  a  system  takes  place,  and  also,  as  in  this  expression  cyclical 
process,  to  designate  a  series  of  changes  in  state,  each  of 
which  takes  place  in  some  definite  manner.)  If  by  such  a 
process  heat  is  to  be  transformed  into  work,  the  Second  Law 
requires  that  an  attendant  change  in  the  condition  of  the 
heat-energy  in  the  surroundings  take  place.  This  change 
always  consists  in  the  passage  of  an  additional  quantity  of 
heat  from  a  state  of  higher,  to  one  of  lower  temperature. 
Difference  of  temperature  in  the  surroundings  is  therefore 


140          GENERAL  PRINCIPLES   OF  PHYSICAL  SCIENCE. 

an  essential  condition  for  the  transformation  of  heat  into 
work  by  a  system  which  itself  undergoes  no  permanent  modi- 
fication ;  otherwise  expressed,  heat  is  not  transformed  into 
work  by  any  cyclical  process  taking  place  in  surroundings  at 
a  constant  temperature.  This  is  merely  a  slightly  different 
statement  of  the  principle  of  the  impossibility  of  perpetual 
motion  of  the  second  kind. 

An  illustration  of  a  simple  cyclical  process  taking  place 
in  surroundings  at  a  constant  temperature,  and  of  the  appli- 
cation to  it  of  the  principle  just  stated  may  be  presented. 
Suppose  that  a  gas  is  allowed  to  expand  against  an  external 
pressure  sensibly  equal  to  or  less  than  its  own,  and  that  it  is 
then,  by  increasing  the  pressure,  brought  back  to  its  original 
volume,  it  being  kept  in  a  large  heat  reservoir  at  constant 
temperature.  The  Second  Law  states  that  the  work  done  by 
the  gas  in  expanding,  even  under  the  most  favorable  condi' 
tions,  can  not  be  greater  than  that  which  must  be  expended 
upon  it  during  the  compression,  since  otherwise  there  would 
be  a  net  gain  of  work  in  a  cyclical  process  at  constant 
temperature. 

It  is  to  be  noted  that  the  fact  that  difference  of  temper- 
ature is  produced  by  the  process  itself  does  not  invalidate  the 
principle  under  consideration ;  for  this  requires  that  if  a  net 
gain  of  work  is  to  result  from  a  cyclical  process,  difference  of 
temperature  must  exist  in  the  surroundings.  Thus,  the  fact 
that  a  gas  in  expanding  rapidly  against  a  pressure  Consider- 
ably less  than  its  own  falls  in  temperature,  and  thereby  gives 
rise  to  a  rapid  flow  of  heat  into  it  from  the  surroundings,  does 
not  increase  the  amount  of  work  produced  by  the  expansion, 
or  make  possible  a  net  gain  of  work  after  the  gas  is  restored 
to  its  original  condition :  on  the  contrary,  as  will  be  seen  be- 
low, it  has  the  opposite  result  —  a  net  loss  of  work.  When 
a  change  in  a  system  takes  place  in  such  a  manner  that  the 
system  itself  does  not  undergo  an  appreciable  variation  of 
temperature,  the  change  and  the  process  by  which  the  change 
is  effected  are  said  to  be  isothermal  ones. 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         141 

The  Second  Law  evidently  permits  that  any  Change  in  a 
system  which  takes  place  of  itself  be  capable  of  producing 
a  definite  maximum  quantity  of  external  work,  namely,  a 
quantity  of  work  which  is  equal  to,  but  not  greater  than,  the 
quantity  which  must  be  expended  in  directly  reversing  the 
change,  and  thereby  restoring  the  system  to  its  original  con- 
dition. And  conversely,  the  Second  Law  requires  that,  in 
order  to  bring  about  a  change  in  a  system  in  the  opposite 
direction  to  that  in  which  it  takes  place  of  itself,  only  a  defi- 
nite minimum  quantity  of  external  work  need  be  expended, 
namely,  a  quantity  of  work  which  is  equal  to,  but  not  less 
than,  the  quantity  which  the  reverse  change  is  of  itself 
capable  of  producing.  It  is,  of  course,  possible,  however, 
that  the  conditions  under  which  the  change  takes  place  or  is 
brought  about,  may  be  such  as  to  produce  less  work  than 
this  maximum,  or  involve  the  expenditure  of  more  work 
than  this  minimum  quantity.  And  discrimination  between 
processes  of  this  kind  and  those  that  involve  the  maximum 
or  minimum  quantities  of  work  is  of  fundamental  importance 
in  applications  of  the  Second  Law. 

To  distinguish  these  two  kinds  of  processes,  it  is  usual 
to  employ  the  terms  irreversible  and  reversible,  which  have, 
however,  primarily,  the  following  significance.  A  change  in 
state,  or  the  process  by  which  it  is  effected,  is  called  reversible, 
when  the  system  can  be  directly  restored  to  its  original  con- 
dition without  giving  rise  to  any  residual  effect  in  the  sur- 
roundings. A  change  in  state  or  a  process  is  irreversible 
when  this  is  not  the  case.  For  example,  if  a  change  in  state 
is  attended  by  the  production  of  as  much  external  work  as 
would  have  to  be  expended  in  restoring  the  system  to  its 
original  condition,  it  is  obviously  reversible ;  but  if  it  pro- 
duces less  work  than  this,  the  change  is  evidently  irreversible, 
for  work  would  have  to  be  withdrawn  from  the  surroundings, 
in  order  to  restore  the  system  to  its  original  state.  Now, 
since  the  Second  Law  does  not  admit  of  the  production 
of  more  work  by  a  change  than  is  required  for  its  reversal, 


142          GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

it  is  evident  that,  when  a  change  in  a  system  is  reversible,  the 
maximum  amount  of  work  is  produced  which  it  is  capable 
of  producing,  or  the  minimum  amount  of  work  is  expendecF" 
by  which  the  change  can  be  brought  about ;  and  that  this 
is  true  only  when  it  is  reversible. 

It  is  important  to  understand  that  the  term  reversible  is 
always  employed,  in  the  manner  just  defined,  to  designate  a 
change  or  process  of  such  a  character  that  it  is  possible  to 
restore  the  original  condition  of  things  both  in  the  system 
and  its  surroundings,  not  in  the  system  alone.  After  an 
irreversible  change  has  taken  place,  it  is  in  general  possible 
to  restore  the  system  to  its  original  condition,  but  only  by 
doing  upon  it  a  larger  quantity  of  work  than  was  obtained 
from  it,  so  that  the  original  condition  in  the  surroundings  is 
not  reproduced.  When  an  irreversible  change  has  once  taken 
place,  it  is  not  possible  by  any  means  whatever  to  reproduce 
in  their  entirety  the  conditions  that  previously  existed. 

Let  us  next  consider  the  conditions  that  must  be  fulfilled, 
in  order  that  an  isothermal  change  may  be  reversible,  or,  what 
is  equivalent  to  this,  in  order  that  it  may  produce  the  max- 
imum amount  of  work.  In  the  first  place,  it  will  be  clear 
from  a  little  consideration  that  the  change  must  take  place 
under  substantially  the  equilibrium-conditions,  the  intensity 
of  any  form  of  energy  that  undergoes  change  within  the  sys- 
tem being  compensated  by  a  sensibly  equal  intensity  exter- 
nally applied ;  for,  since  the  work  produced  by  a  change  will 
be  greater,  the  greater  the  magnitude  of  the  latter  intensity, 
this  must  be  as  great  as  is  consistent  with  the  occurrence  of 
the  change :  the  change  will  occur,  however,  if  the  external 
intensity  is  less  than  that  within  the  system  by  only  an  in- 
finitesimal amount.  Similarly,  since  the  work  expended  in 
producing  the  reverse  change  will  be  smaller,  the  smaller  the 
value  of  the  externally  applied  intensity,  this  must  be  as  small 
as  is  consistent  with  the  production  of  the  change :  the  change 
can  be  brought  about,  however,  by  the  external  application  of 
an  intensity  greater  by  only  an  infinitesimal  amount  than 
that  corresponding  to  the  equilibrium-condition. 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         143 

Consider,  as  an  example,  a  gas  within  a  cylinder  closed 
at  the  top  by  a  piston  on  which  a  weight,  producing  any 
desired  downward  pressure,  can  be  placed,  the  piston  being 
held  in  position  by  a  suitable  stop  by  which  it  can  be  released 
when  desired.  It  is  evident,  now,  that  the  work  that  will  be 
done  if  the  gas  expands  will  be  greater,  the  greater  the  weight 
upon  the  piston ;  but  that  the  gas  will  not  expand  at  all,  if 
the  weight  on  the  piston  gives  rise  to  a  greater  downward 
pressure  than  the  upward  pressure  of  gas  beneath.  The 
maximum  amount  of  work  that  can  be  obtained  by  the  ex- 
pansion will  therefore  be  obtained  when  the  externally  applied 
pressure  is  sensibly  equal  to  that  of  the  gas.  (Since  that  of 
the  gas  continually  diminishes  as  its  volume  increases,  the 
weight  on  the  piston  must  of  course  be  correspondingly 
decreased  as  the  expansion  progresses.)  Similarly,  the  ex- 
ternal pressure  that  must  be  applied  in  compressing  the  gas 
to  its  original  volume,  if  the  minimum  amount  of  work  is  to 
be  expended,  must  be  greater  than  that  of  the  gas  by  only 
an  infinitesimal  amount.  It  is  further  evident,  since  the 
pressure  during  the  compression  need  be  greater  than  that 
during  the  expansion  by  only  an  infinitesimal  amount,  that 
the  amounts  of  work  involved  are  sensibly  equal  when 
the  two  opposing  changes  take  place  under  the  equilibrium- 
conditions,  and  that  therefore  each  of  them  is  then  reversible, 
since  the  gas  can  be  restored  to  its  original  volume  without 
producing  any  permanent  change  in  the  surroundings. 

As  another  example,  consider  the  conditions  for  the  pro- 
duction of  the  maximum  quantity  of  external  work  from  a 
chemical  change,  such  as  that  which  takes  place  between 
a  zinc  plate  and  a  copper  sulphate  solution.  If  the  zinc  is 
placed  directly  in  the  solution,  the  change  is  not  attended 
by  the  production  of  any  work  whatever,  and  it  is  of  course 
completely  irreversible.  If,  however,  the  plate  of  zinc  is 
placed  in  a  zinc  sulphate  solution  and  a  plate  of  copper 
is  placed  in  a  copper  sulphate  solution  that  is  in  contact 
with  the  zinc  sulphate  solution,  it  is  found  that  a  potential- 


144          GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

difference  exists  at  the  two  plates,  and  that  by  suitably 
connecting  them  outside  the  cell,  electrical  energy,  and 
secondarily  other  forms  of  energy,  can  be  produced.  If  the 
plates  are  connected  only  by  an  ordinary  metallic  conductor, 
it  is  still  true,  however,  that  the  only  result  is  the  production 
of  a  quantity  of  heat  in  the  conductor  owing  to  its  electrical 
resistance,  and  the  change  is  completely  irreversible,  just  as 
in  the  former  case,  the  only  difference  being  that  the  heat 
is  now  produced  in  the  surroundings  instead  of  in  the  react- 
ing mixture.  But  if  the  electromotive  force  of  the  cell  be 
compensated  by  placing  in  the  external  circuit  a  storage  cell 
or  electric  motor  exerting  an  opposite  electromotive  force 
infmitesimaHy  less  than  that  of  the  cell,  it  is  evident  that  the 
change  then  takes  place  under  the  equilibrium-conditions, 
and  that  it  produces  the  maximum'  amount  of  work  in  the 
surroundings  ;  for,  if  the  counter  electromotive  force  were  not 
less  than  that  of  the  cell  by  an  infinitesimal  amount,  the 
change  would  not  take  place  at  all,  and,  if  it  were  -less  by 
a  finite  amount,  a  finite  current  would  result,  which  would 
be  attended  by  a  finite  Joule  Heat-Effect  and  an  equivalent 
decrease  in  the  work  produced.  It  is  evident,  moreover,  that 
the  change  is  then  reversible;  for,  if  the  external  electro- 
motive force  be  made  greater  than  that  of  the  cell  by  only 
an  infinitesimal  amount,  electricity  will  flow  in  the  opposite 
direction,  the  chemical  change  in  the  cell  will  take  place  in 
the  reverse  sense  (zinc  being  precipitated  and  copper  dis- 
solved), and  the  quantity  of  electrical  energy  thereby  expended 
in  restoring  the  cell  to  its  original  state  will  be  greater  than 
that  obtained  from  it  by  only  an  infinitesimal  amount. 

Another  condition  which  must  obviously  be  fulfilled,  in 
order  that  a  change  may  take  place  reversibly,  is  that  the 
process  which  is  essential  to  the  production  of  the  change, 
and  which  takes  place  under  the  equilibrium-conditions,  be  not 
attended  by  any  other,  irreversible  process,  that  is,  by  any 
process  by  which  work  is  converted  into  heat  without  pro- 
ducing such  a  change  in  the  system  or  surroundings  that  the 


GENERAL  PRINCIPLES  RELATING   TO   ENERGY.         145 

work  can  be  again  obtained  from  the  heat.  Two  such  irre- 
versible processes  of  extremely  common  occurrence  are  the 
production  of  heat  by  friction  and  by  electrical  resistance. 
Thus,  in  a  reversible  expansion  or  compression  of  a  gas,  the 
piston  must  move  in  the  cylinder  without  friction ;  or,  in  a 
reversible  transformation  of  the  electrical  energy  produced 
by  a  voltaic  battery  into  mechanical  energy  by  means  of  an 
electric  motor,  the  Joule  Heat-Effect,  and  hence  either  the 
current-strength  or  the  resistance  of  the  circuit,  must  be  zero. 

From  the  characteristics  of  reversible  changes  just  con- 
sidered, it  will  be  seen  that  no  actually  occurring  changes  are 
completely  reversible  ;  for,  in  order  that  a  change  may  be 
such,  it  is  necessary  that  there  be  only  an  infinitesimal  dif- 
ference of  intensity  within  and  without  the  system,  and  when 
this  is  the  case,  the  time  required  for  the  occurrence  of  the 
change  becomes  infinite.  •  Moreover,  it  is  never  possible  to 
eliminate  entirely  such  irreversible  processes  as  the  produc- 
tion of  heat  by  friction  or  by  electrical  resistance.  Reversible 
-  changes  are  therefore  ideal  changes  representing  a  limiting 
condition  which  may  be  approached  more  or  less  closely  in 
the  case  of  actual  changes,  but  is  never  attained  by  them. 
Since  all  actual  changes  are  to  a  greater  or  less  extent  irre- 
versible, the  proportion  of  the  existing  energy  that  is  trans- 
formable into  work  is  continually  diminishing. 

The  fact  that  reversible  changes  and  processes  consti- 
tute a  limiting  case,  and  one  which  involves  the  equilibrium- 
conditions  of  systems,  makes  the  consideration  of  them  of 
the  greatest  scientific  value.  It  is  therefore  especially  impor- 
tant to  state  explicitly  the  forms  of  the  Second  Law  which 
are  applicable  to  them.  It  is  clear  that  the  following  prin- 
ciple in  regard  to  them  is  a  necessary  consequence  of  the 
general  statement  of  the  Second  Law  and  of  the  definition 
of  reversible  processes :  the  algebraic  sum  of  all  the  quanti- 
ties of  work  produced  in  any  isothermal  reversible  cyclical 
process  is  equal  to  zero ;  for  if  the  sum  were  greater  than 
zero,  the  process  itself,  and  if  it  were  less  than  zero,  the  re- 


146          GENERAL  PRINCIPLES   OF  PHYSICAL  SCIENCE. 

versal  of  the  process,  would  be  a  contradiction  of  the  Second 
Law.  This  principle  is  essentially  identical  with  the  following 
one  :  the  quantity  of  external  work  produced  when  a  definite 
change  in  the  state  of  a  system  takes  place  at  constant  tem- 
perature is  independent  of  the  process  by  which  the  change 
is  effected,  provided  only  that  it  be  a  reversible  one ;  for  if 
two  reversible  processes  of  bringing  about  the  same  change 
in  state  produced  different  quantities  of  work,  they  could 
obviously  be  combined  into  a  cyclical  process  yielding  a 
quantity  of  work. 

By  the  application  of  this  principle  many  important 
conclusions  in  regard  to  the  equilibrium-conditions  of  various 
systems  and  many  relationships  between  the  intensity-factors 
of  different  forms  of  energy  within  them  have  been  derived. 
One  of  the  common  methods  of  deriving  such  conclusions 
and  relationships  is  to  consider  an  isothermal  reversible  cycli- 
cal process  involving  different  forms  of  energy,  or  the  same 
form  under  different  conditions,  to  be  carried  out,  to  calculate 
the  quantities  of  work  involved  in  the  different  parts  of  the 
process,  and  to  place  their  sum  equal  to  zero,  in  accordance 
with  the  principle  just  stated. 

This  method  may  be  illustrated  by  the  following  example, 
in  which  two  different  forms  of  energy  are  involved.  Con- 
sider a  system  (a  voltaic  cell)  composed  of  two  platinum 
electrodes  coated  with  platinum-black,  each  of  which  is  half 
immersed  in  dilute  sulphuric  acid,  the  other  half  being  ex- 
posed to  hydrogen  gas,  which  is  contained  at  different  press- 
ures, /!  and  /2,  in  two  vessels  of  infinite  capacity  surround- 
ing the  upper  parts  of  the  two  electrodes.  An  electromotive 
force  is  found  to  exist  at  the  electrodes.  Consider  these  to 
be  connected  with  each  other  by  a  conducting  medium  of 
infinitesimal  resistance,  and  that  a  counter  electromotive  force 
is  introduced  into  the  circuit.  By  this  system  an  electric 
current,  and  therefore  electrical  energy,  can  be  produced. 
The  processes  which  attend  this  production  of  work  are  the 
absorption  of  hydrogen  by  the  platinum  electrode  which  is  in 


GENERAL   PRINCIPLES  RELATING   TO  ENERGY.         147 

contact  with  the  more  compressed  gas,  the  passage  of  this 
hydrogen  from  the  electrode  into  solution,  its  precipitation  on 
the  other  electrode,  and  its  escape  therefrom  into  the  space 
containing  the  less  compressed  gas,  the  net  change  in  the 
system  therefore  consisting  in  a  transfer  of  some  of  the 
hydrogen  from  the  vessel  containing  it  at  the  higher  to  that 
containing  it  at  the  lower  pressure.  Moreover,  it  is  found 
that  the  system  can  be  restored  to  its  original  condition,  that 
is,  the  process  can  be  made  to  take  place  in  the  opposite 
direction  and  the  hydrogen  be  brought  back  to  the  higher 
pressure,  by  making  the  counter  electromotive  force  greater 
by  an  infinitesimal  amount  than  that  of  the  cell.  It  is  there- 
fore possible  to  carry  out  the  process  reversibly.  In  order  now 
to  determine  how  much  electrical  energy  can  be  produced  by 
a  definite  change  in  the  system,  and  what  the  electromotive 
force  of  the  cell  is,  we  consider  a  process  consisting  of  two 
parts  to  be  carried  out :  i.  We  transfer  1.0075  grams  (one 
equivalent  weight)  of  hydrogen  from  the  vessel  containing  it 
at  the  higher  pressure  (p^  to  that  containing  it  at  the  lower 
pressure  (/2)  by  means  of  the  voltaic  cell  just  described, 
the  potential-difference  of  the  cell  being  compensated  by  an 
opposite  and  sensibly  equal  potential-difference  externally 
applied,  for  example,  by  placing  in  series  with  it  a  motor 
running  at  a  suitable  speed,  so  that  the  maximum  amount  of 
work  will  be  obtained  from  it.  2.  We  compress  the  1.0075 
grams  of  hydrogen  obtained  in  the  vessel  at  lower  pressure, 
with  the  help  of  a  cylinder  provided  with  a  frictionless  piston, 
until  its  pressure  /2  is  brought  back  to  its  original  value  /x. 
In  order  to  do  this  reversibly,  or  with  the  expenditure  of  the 
minimum  amount  of  work,  we  take  care  that  the  external 
pressure  upon  the  piston  be  always  greater  than  that  of 
the  gas  beneath  by  only  an  infinitesimal  amount.  During 
both  parts  of  the  process  we  keep  the  system  in  a  large  heat 
reservoir  of  infinite  extent,  so  that  no  difference  of  temper- 
ature is  produced.  An  isothermal  reversible  cyclical  process 
has  now  been  carried  out,  and  the  Second  Law  requires  that 


KAPfl  MIM    LIBRARY 

CHEM.  BLDG.    u.  c 


148         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

the  sum  of  the  quantities  of  work,  Wf  and  WIh  produced  in 
the  two  parts  of  it,  be  equal  to  zero.  That  is,  since 

Wf  =  EQ  (by  §*32),  and  Wn  =  NRT  log(/2  /.fr)  (by  §  30), 
it  follows  that,  Wt  +  WH  =  EQ  +  NR  T  log(/2  /  /x)  =  o, 
or  that,  E  Q  =  NR  T  log(/i  /  /2) . 

In  this  expression  s  signifies  the  electromotive  force  of  the 
cell,  Q  the  quantity  of  electricity  produced  when  one  equiva- 
lent weight  of  hydrogen  is  transferred,  N  the  number  of  mols 
of  hydrogen  transferred  —  in  this  case  0.5,  since  one  mol  of 
hydrogen  is  2.015  grams  or  two  equivalents,  T  the  absolute 
temperature  at  which  the  change  takes  place,  and  R  the  gas- 
constant,  which  must  be  expressed  in  joules,  if  the  electrical 
energy  is  to  be  so  expressed,  as  is  usually  the  case.  In  order 
to  determine  the  value  of  the  electromotive  force  E,  additional 
knowledge  is  obviously  required.  With  the  help  of  Faraday's 
Law  and  the  electrochemical  constant  (§33)  however,  the  cal- 
culation can  be  readily  made  ;  for  these  require  that,  when  one 
equivalent  weight  of  hydrogen  (or  of  any  other  substance) 
passes  into  solution  or  is  precipitated  in  a  voltaic  cell,  96600 
coulombs  of  electricity  be  under  all  circumstances  developed. 
Thus,  substituting  the  appropriate  values  in  the  above  equa- 
tktfv,  we  get,  for  a  temperature  of  20°  : 

96600  X  E  =  0.5  X  8.31  X  293  X  2.303  log10(A/A)  volts. 

If  the  pressure  in  one  vessel  were  two  atmospheres,  and  that 
in  the  other,  one  atmosphere,  (pi  //2)  =  2,  E  =  0.00874  volts, 
and  Wf  =  EQ=  843.8  joules. 

40.  Application  of  the  Second  Law  to  Changes 
Taking  Place  at  Different  Temperatures.  —  It  has  been 
already  seen  that,  when  a  quantity  of  heat  is  transformed  into 
work  by  a  system  which  undergoes  no  permanent  modifi- 
cation, an  additional  quantity  of  heat  is  always  taken  up 
from  surroundings  at  a  higher  temperature  and  is  transferred 
to  surroundings  at  a  lower  temperature.  That  is  to  say,  even 
when  a  difference  of  temperature  exists,  only  a  fraction  of 


GENERAL   PRINCIPLES  RELATING   TO  ENERGY.         149 

the  heat  taken  up  by  the  system  from  the  warmer  surround- 
ings can  be  transformed  into  work,  if  the  system  is  to  be 
restored  to  its  original  condition.  Important  questions  at 
once  arise  as  to  what  determines  the  fraction  that  can  be 
transformed :  Is  it  dependent  on  the  nature  of  the  system 
employed  ?  And  how  does  it  vary  with  the  temperatures  ? 
To  the  consideration  of  these  questions  and  to  applications 
of  the  conclusions  reached  this  section  will  be  devoted. 

It  should  be  first  pointed  out,  however,  if  the  maximum 
amount  of  work  is  to  be  obtained  by  a  process  taking  place 
at  different  temperatures,  that  not  only  the  two  conditions 
mentioned  in  §  39  must  be  fulfilled,  namely,  the  occurrence  of 
all  changes  under  the  equilibrium-conditions  and  the  absence 
of  any  attendant  essentially  irreversible  process,  such  as  fric- 
tion or  electrical  resistance  ;  but  also  that  the  changes  in  state 
of  the  system  must  not  be  attended  by  the  direct  passage  of 
any  quantity  of  heat  from  the  surroundings  at  the  higher, 
to  those  at  the  lower  temperature  ;  for  example,  no  heat 
must  be  transferred  by  processes  of  conduction  or  radiation. 
For  a  quantity  of  heat  so  passing  does  not  produce  any  work, 
while  under  appropriate  '  conditions  a  certain  fraction  of  that 
quantity  of  heat,  as  of  any  other  quantity,  can,  of  course,  be 
transformed  into  work.  This  third  condition  is,  like  the  other 
two,  comprehended  in  the  statement  that  a  process  produces 
the  maximum  amount  of  work  only  when  it  is  reversible. 
For  the  direct  flow  of  heat  from  a  higher  to  a  lower  tem- 
perature is  essentially  an  irreversible  process :  the  heat  will 
not,  of  itself,  pass  back  to  the  body  at  the  higher  temper- 
ature, and  can  not  be  brought  there  except  by  the  expendi- 
ture of  work,  that  is,  without  producing  a  permanent  change 
in  the  surroundings. 

In  order  now  to  determine  whether  the  value  of  the 
ratio  of  the  quantity  of  heat  that  can  be  transformed  into 
work  to  the  quantity  transferred  from  the  higher  to  the  lower 
temperature  is  dependent  upon  the  nature  of  the  system 
employed  for  the  transformation,  or  upon  the  way  in  which 


160         GENERAL   PRINCIPLES  OF  PHYSICAL  SCIENCE. 

the  transformation  is  carried  out,  let  us  assume  that  two 
different  reversible  cyclical  processes,  carried  out  with  differ- 
ent systems  or  in  a  different  way  with  the  same  system,  could 
produce  two  unequal  quantities  of  work  by  transferring  an 
equal  quantity  of  heat  from  a  higher  to  a  lower  temperature ; 
.suppose  now  we  cause  the  process  that  produces  the  larger 
amount  of  work  ( W")  to  take-  place  in  such  a  way  that 
it  takes  up  a  quantity  of  heat  (Qi)  at  the  higher  tempera- 
ture (7\),  transfers  a  part  of  it  (Q2)  to  the  lower  temperature 
(T2),  and  transforms  the  remainder  into  work  ( W") ;  and 
.suppose  we  cause  the  other  process,  which  in  transferring 
the  same  quantity  of  heat  (Q2)  from  7i  to  Tz  produces  the 
smaller  amount  of  work  (  W),  to  take  place  in  the  reverse 
direction  —  that  is,  so  that  it  takes  up  the  heat  (Q2)  trans- 
ferred by  the  former  process  to  the  lower  temperature,  and 
raises  it  to  the  higher  temperature  by  expending  the  required 
.amount  of  work  ( W).  It  is  then  evident  that  the  net  result 
of  these  operations  would  be  the  production  of  a  quantity  of 
work  ( W"  —  Wr)  from  an  equivalent  quantity  of  heat  with- 
out any  other  permanent  change  whatever  having  been 
brought  about  either  in  the  systems  or  in  the  surroundings. 
Since  this  is  contrary  to  the  fundamental  statement  of  the 
Second  Law,  the  supposition  made  that  the  two  processes 
produce  unequal  quantities  of  work  is  untenable.  That  is  to 
.say,  the  quantity  of  work  produced  from  a  definite  quantity 
of  heat  by  any  reversible  cyclical  process  taking  place  at 
definite  temperatures  is  independent  of  the  system  employed 
and  of  the  way  in  which  the  process  is  carried  out.  Or,  since 
a  reversible  process  produces  the  maximum  amount  of  work, 
and  since  in  a  cyclical  process  the  system  is  not  permanently 
modified,  this  principle  can  be  stated,  without  using  these 
special  terms,  as  follows :  The  maximtim  amount  of  work 
which  can  be  obtained  when  a  definite  quantity  of  heat  is 
transferred  front  one  temperature  to  another  by  any  process 
in  which  the  system  employed  undergoes  no  permanent  mod- 
ification is  not  dependent  on  the  nature  of  the  system  or  of 
.the  process. 


GENERAL  PRINCIPLES  RELATING    TO  'ENERGY.         151 

By  the  conclusion  just  reached  the  determination  of  the 
relation  between  temperature  and  the  proportion  of  heat  trans- 
formable into  work  is  greatly  facilitated ;  for  evidently  it  is 
now  only  necessary  to  determine  what  that  relation  is  for  a 
single  reversible  cyclical  process.  We  will  therefore  consider 
a  process  in  which  a  definite  quantity  (N  mols)  of  a  perfect 
gas  contained  in  a  vessel  closed  with  a  weighted  frictionless. 
piston  is  employed  as  the  transforming  system,  and  we  will 
assume  that  the  temperatures  involved  are  defined  to  be 
proportional  to  its  pressures,  instead  of  to  those  of  hydrogen,, 
as  was  done  in  §  22. 

Suppose  that  the  gas  has  in  the  beginning  a  volume  v± 
and  temperature  7*1,  and  that  the  following  process,  consisting 
of  four  distinct  parts,  is  carried  out  with  it : 

First,  place  it  in  a  large  heat-reservoir  (for  example,  a 
large  water-reservoir)  at  the  temperature  Tlt  and,  by  gradu- 
ally diminishing  the  weight  on  the  piston,  cause  the  gas  to 
expand  slowly  until  its  volume  becomes  vz.  The  gas  does 
work  during  the  expansion  and  tends  to  cool  off,  but  is  kept 
at  constant  temperature  by  an  absorption  of  heat  from  the 
reservoir. 

Second,  make  the  piston  immovable,  so  that  the  volume 
must  remain  constant,  and  withdraw  heat  from  the  gas  until 
its  temperature  falls  from  7\  to  T2 ;  in  order  to  do  this  rever- 
sibly,  the  gas  is  brought  successively  into  communication 
with  a  series  of  heat-reservoirs  varying  continuously  in  tem- 
perature from  7*!  to  72,  in  such  a  manner  that  the  gas  always 
differs  in  temperature  by  only  an  infinitesimal  amount  from 
the  reservoir  with  which  it  is  in  contact.  The  heat  which 
the  gas  gives  out  in  cooling,  instead  of  all  falling  in  tempera- 
ture from  7i  to  72,  as  would  be  the  case  if  the  gas  were  trans- 
ferred directly  to  the  heat-reservoir  at  T2,  is  thus  distributed 
among  the  reservoirs  at  the  intermediate  temperatures,  and 
would  evidently  serve  to  heat  the  gas  to  its  original  tempera- 
ture, whence  it  follows  that  the  process  is  reversible. 


152          GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

Third,  place  the  gas  in  a  large  heat-reservoir  at  the  tem- 
perature T2,  and  slowly  compress  it  by  releasing  the  piston 
and  gradually  increasing  the  weight  upon  it  until  the  volume 
of  the  gas  has  been  reduced  from  va  to  its  original  volume  vlf 
Work  is  done  on  the  gas  during  the  compression,  and  it  tends 
to  become  heated  ;  but  if  is  kept  at  constant  temperature  by 
the  reservoir  with  which  it  is  surrounded,  which  absorbs  the 
heat  produced. 

Fourth,  again  make  the  piston  immovable,  so  as  to  keep 
the  volume  constant,  and  impart  heat  to  the  gas  until  its  tem- 
perature rises  from  T2  to  7\,  making  use  of  the  series  of  res- 
ervoirs of  continuously  varying  temperature  employed  in  the 
second  part  of  the  process. 

It  is  evident  that  the  gas  is  now  in  its  original  condition, 
and  that  each  of  the  changes  which  it  has  gone  through  has 
taken  place  in  such  a  manner  that  it  could  be  reversed  with- 
out giving  rise  to  any  residual  effect  in  the  surroundings  ;  or, 
in  other  words,  that  a  reversible  cyclical  process  has  been 
carried  out. 

Let  us  now  proceed  to  determine  the  quantities  of  heat 
and  work  involved  in  the  separate  parts  of  this  process,  and 
from  these,  the  net  result  of  the  process  with  respect  to  the 
transformation  of  heat  into  work.  In  the  first  place,  it  is 
clear,  since  the  volume  of  the  gas  was  kept  constant,  that  no 
work  was  done  in  the  second  and  fourth  parts  of  the  process ; 
and  furthermore,  since  the  heat-capacity  of  a  definite  quan- 
tity of  a  perfect  gas  is  independent  of  its  pressure  and  vol- 
ume (§  37),  that  the  heat  given  out  by  the  gas  to  the  reser- 
voirs in  the  second  part  of  the  process  is  equal  to  that  taken 
up  by  it  from  them  in  the  fourth  part.  Therefore,  since  no 
heat  is  transformed  into  work  nor  transferred  from  one  tem- 
perature to  another  as  a  result  of  these  two  parts  of  the  pro- 
cess, they  can  be  entirely  left  out  of  consideration. 

It  remains  only  to  consider  the  first  and  third  parts  of 
the  process.  Designate  the  work  produced  in  these  two  parts 
by  Wl  and  WZt  respectively,  and  the  heat  absorbed  from  the 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.        153 

surroundings  in  the  two  parts  by  Qi  and  <22>  respectively.     It 
follows  then  by  §  30  that  : 

Wi  =  NR  71  log  ^L,  and    W*  =  NR  Tz  log  ^L, 

V-i  Vi 

and  therefore  that   the  quantity  of  work  produced  by  the 
whole  process, 

W=  W,+  W,  =  (71  -  T*)NR  log  ^1. 


Tt  follows,  furthermore,  from  the  principle  that  the  heat 
absorbed  by  a  perfect  gas  in  expanding  at  a  constant  tem- 
perature is  equivalent  to  the  work  done  by  it  (§  37),  that  : 

Ql=  Wi  = 


Dividing  the  former  of  these  two  equations  by  the  latter,  we 
get  an  expression  for  the  amount  of  work  produced  in  the 
process  above  described,  in  terms  of  the  quantity  of  heat 
absorbed  from  the  surroundings  at  the  higher  temperature  and 

T  —  —  T* 

the  two  temperatures  involved  ;  that  is  :    W  =  Ql  -1  -  ?. 

7i 

Since  it  has  been  shown  above  that  the  Second  Law 
requires  that  the  same  quantity  of  work  be  produced  from 
a  definite  quantity  of  heat  by  any  reversible  cyclical  process 
whatever  taking  place  at  the  same  two  temperatures,  it  is 
evident  that  this  equation  is  an  entirely  general  expression 
of  the  application  of  the  Second  Law  to  such  processes. 
Thus,  just  the  same  final  result  is  reached  from  the  consider- 
ation of  the  quantities  of  work,  and  heat  involved  in  the 
separate  parts  of  the  process  known  as  Carnot's  Cyclical 
Process,  in  which  a  perfect  gas  undergoes  a  series  of  changes 
in  pressure,  volume,  and  temperature  of  a  different  character 
from  those  specified  above  :  this  process  will  be  found  de- 
scribed in  treatises  on  Heat  and  Thermodynamics.  More- 
over, the  ratio  of  work  produced  to  heat  absorbed  would  be 
the  same  if  the  system  employed  in  the  process  were  any 
other  than  a  perfect  gas  ;  thus,  this  would  be  true  in  the  case 


154         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

of  cyclical  processes  involving  changes  in  the  concentration 
and  temperature  of  dilute  solutions,  which  we  now  know  how, 
at  least  in  principle,  to  carry  out  reversibly. 

The  conclusion  reached  may  be  stated  as  follows :  The 
quantity  of  work  produced  in  any  reversible  cyclical  process 
in  which  a  quantity  of  heat  is  absorbed  by  a  system  from 
surroundings  at  one  temperature,  and  the  portion  of  it  not 
transformed  into  work  is  given  off  to  surroundings  at  an- 
other temperature,  is  equal  to  the  heat  absorbed  at  the  higher 
temperature  multiplied  by  the  difference  of  temperature  and 
divided  by  the  higher  temperature  expressed  on  the  absolute 
scale.  If  the  process  is  not  reversible,  a  less  amount  of  work 
is  produced. 

Therefore,  the  maximum  quantity  of  work  ( W)  that  can 
be  obtained  from  a  definite  quantity  of  heat  (<2i)  at  any 
definite  temperature  (7i)  with  the  help  of  any  system  that 
undergoes  no  permanent  modification  is  directly  proportional 
to  the  difference  between  that  temperature  and  any  lower 
temperature  (Tz)  with  which  the  system  can  be  brought  into 
communication.  If  the  system  can  not  be  exposed  to  sur- 
roundings of  different  temperatures,  no  work  can  be  produced 
by  it,  as  was  concluded  in  §  39  ;  for,  when  T2  =  7i,  W  =  o. 
If,  on  the  other  hand,  it  were  practicable  to  bring  the  system 
into  communication  with  surroundings  at  the  absolute  zero  of 
temperature,  it  would  possess  the  power  of  transforming  the 
heat  of  surroundings  at  any  higher  temperature  completely 
into  work ;  for,  when  Jjj  =  o,  W  =  Q±.  It  will  thus  be  seen 
that  the  general  statement  of  the  Second  Law  postulated  at 
the  start  would  not  be  true,  if  surroundings  of  great  heat- 
capacity  at  the  absolute  zero  of  temperature  were  accessible  to 
us  ;  but  such  surroundings  are  not  accessible.  It  is  further 
"to  be  noted  that  the  maximum  amount  of  work  obtainable 
from  a  definite  quantity  of  heat,  in  different  cases  in  which 
an  equal  difference  of  temperature  exists  in  the  surroundings, 
is  inversely  proportional  to  the  higher  temperature ;  that  is,  if 
(7\  —  r2)  is  constant,  W*  (l  /  7\).  Thus,  a  reversible  cycli- 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         156 

cal  process  taking  place  at  100°  and  o°  (on  the  normal  tem- 
perature scale)  would  produce  only  |||  as  much  work  from  a 
definite  quantity  of  heat  at  100°  as  one  taking  place  at  o° 
and — 1 00°  would  produce  from  the  same  quantity  of  heat 
at  o°. 

The  temperatures  T±  and  T2  in  the  equation  are  those 
that  would  be  measured  by  means  of  a  thermometer  contain- 
ing a  constant  volume  of  a  perfect  gas,  instead  of  one  of 
hydrogen  as  has  been  previously  understood  in  accordance 
with  the  definition  of  §22.  But,  since  the  experiments  of 
Joule  and  Thomson  show  that  in  the  case  of  hydrogen  at 
temperatures  between  the  freezing  and  boiling  points  of  water 
the  change  in  internal  energy  upon  expansion  is  extremely 
small,  the  two  temperature-scales  are  almost  identical  between 
those  two  temperatures.  In  fact,  computations  based  on  the 
data  of  the  porous-plug  experiments  have  shown,  assuming 
the  interval  between  the  freezing  and  boiling  points  of  water 
to  be  1 00°  on  each  scale,  that  the  intermediate  temperatures 
would  not  at  any  point  differ  by  more  than  0.003°,  and  tnat 
the  calculated  temperature  of  the  freezing-point  above  the 
absolute  zero  would  be  within  0.1°  of  273°,  whether  a  hydro- 
gen thermometer  or  a  perfect-gas  thermometer  were  employed 
for  the  measurements.  It  is  not  improbable,  however,  that 
the  two  scales  differ  to  a  considerable  extent  at  very  low 
temperatures. 

It  is  also  to  be  noted  that  a  new  definition  of  temper- 
ature —  one  that  is  not  based  on  any  property  of  any  definite 
kind  of  substance,  but  merely  on  the  characteristics  of  heat 
energy  —  can  be  founded  upon  the  application  of  the  Second 
Law  to  reversible  cyclical  processes  of  the  kind  just  con- 
sidered. The  result  reached  can  be  written  in  the  form, 
Qi/(—Q*)=TJ  Tz,  since  by  the  First  Law  W=  Ql  +  &•; 
and,  in  accordance  with  this  equation,  we  may  define  the  ratio 
of  any  two  temperatures  to  be  equal  to  the  ratio  of  the  heat 
absorbed  at  the  one  temperature  to  the  heat  evolved  at  the 
other  temperature,  when  heat  is  transferred  from  the  one  to 


153         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

the  other  by  any  reversible  cyclical  process  whatever.  The 
temperature-scale  based  upon  this  definition  is  called  the 
absolute  energetic  (or  thermo dynamic)  temperature-scale.  It 
is  absolute  in  the  proper  sense  of  the  word ;  for,  unlike  the 
gas  or  mercury  thermometer  scales,  its  definition  does  not 
involve  a  relation  to  any  property  of  any  definite  substance. 
Temperatures  expressed  on  this  scale  are,  of  course,  exactly 
identical  with  those  that  are  proportional  to  the  pressures 
that  would  be  exhibited  by  a  constant-volume  thermometer 
filled  with  a  perfect  gas. 

The  principle  derived  above,  which  is  expressed  by  the 
equation,  W=  Ql  (7i  —  T2)  /  T1}  is  especially  important  by 
reason  of  the  fact  that  it  can  be  applied  to  determining  the 
effect  of  temperature  on  the  equilibrium-conditions  of  systems, 
or  on  the  value  of  the  intensity-factor  of  any  form  of  energy 
that  may  be  involved.  An  illustration  of  the  way  in  which 
the  principle  is  applied  in  obtaining  such  results  may  there- 
fore be  presented.  Consider  a  reversible  cyclical  process 
carried  out  at  two  different  temperatures  with  the  hydrogen- 
gas-cell  described  in  a  previous  example  (§  39).  Let  the 
process  consist  of  the  following  parts  :  i .  Cause  one  equiva- 
lent of  hydrogen  to  pass,  with  the  help  of  the  cell  operat- 
ing reversibly  at  the  temperature  7\,  from  the  vessel  contain- 
ing it  at  the  higher  pressure  /1?  to  that  containing  it  at  the 
lower  pressure /2.  2.  Cool  the  gas  so  obtained  by  means  of 
a  series  of  heat  reservoirs  to  a  lower  temperature  7^,  keeping 
its  volume  constant,  its  pressure  being  thereby  reduced  to  /8. 
3.  By  means  of  the  voltaic  cell,  operating  reversibly  at  the 
temperature  7*2,  transfer  the  hydrogen  from  a  vessel  in 
which  the  pressure  is  /8  to  one  in  which  it  has  a  higher  value 
/4,  this  last  pressure  being  such  that  the  gas  will  return  to  its 
original  pressure  p19  if  heated  at  constant  volume  to  the  tem- 
perature 7i.  4.  Heat  the  gas  so  obtained  by  means  of  the 
series  of  heat  reservoirs  to  the  original  temperature  7i,  keep- 
ing its  volume  constant.  The  reversible  cyclical  process  is 
then  complete.  As  no  work  was  done  in  the  second  and 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         157 

fourth  parts  of  the  process,  and  as  the  heat  given  out  by  the 
gas  in  the  second  part  was  exactly  compensated  by  that  taken 
up  by  it  in  the  fourth  part,  since  the  heat-capacity  of  a 
quantity  of  gas  is  independent  of  its  pressure,  these  parts  of 
the  process  can  be  entirely  left  out  of  consideration.  There- 
fore, only  the  quantities  of  work  and  heat  involved  in  the  first 
and  third  parts  need  to  be  considered.  If  the  electromotive 
force  of  the  cell  at  7\  is  E±  and  at  Tz  is  E^  and  the  quantity 
of  electricity,  which  by  Faraday's  Law  is  the  same  in  both 
cases,  is  Q  ,  the  work  produced  in  the  first  part  of  the  process 
is  EI  <2,  and  that  expended  in  the  third  part  is  EZ  Q  /  the  work 
W  produced  by  the  whole  process  is,  therefore,  (EI  —  EZ)Q. 
Substituting  this  value  in  the  general  equation,  we  get : 

(E,  —  EZ)Q  =  Q,  (7\  —  Tt)  /  7\, 

where  Q\  is  the  heat  absorbed  in  the  first  part  of  the  process. 
This  equation  evidently  expresses  a  relation  between  the 
maximum  quantities  of  electrical  work  which  the  cell  is 
capable  of  producing  at  two  different  temperatures,  or  be- 
tween the  values  of  its  electromotive  force  at  two  different 
temperatures,  in  terms  of  the  heat-effect  that  attends  the 
operation  of  the  cell  under  the  equilibrium-conditions  at  one 
of  the  two  temperatures.  This  result  is  a  typical  one,  being 
entirely  analogous  to  those  reached  by  other  similar  applica- 
tions of  the  Second  Law.  In  this  special  case,  the  work, 
Wi  =  EIQ,  produced  in  the  first  part  of  the  process  can  be 
substituted  for  the  heat  Qi  absorbed  in  it ;  for  the  two  quanti- 
ties are  in  this  case  equivalent,  since  a  gas  does  not  change 
its  own  energy-content  when  changed  in  pressure  and  volume 
at  a  constant  temperature.  Making  this  substitution,  the  last 
equation  becomes  simplified  to  the  following  one  : 

(*i  —  *,)  /  E,  =  (71  —  Tt)  I  7\;  or  E,  /  E,  =  7\  /  T, ; 

which  states  that  the  fractional  change  in  the  electromotive 
force  of  the  cell  is  equal  to  the  fractional  change  in  its 
absolute  temperature,  or  that  its  electromotive  force  is  pro- 
portional to  its  absolute  temperature. 


158         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

It  is  important  to  note  that  the  above  equation  will  yield 
exact  results  only  when  applied  to  reversible  cyclical  processes 
in  which,  as  in  those  used  for  its  derivation  and  illustration, 
quantities  of  heat  are  absorbed  at  one  temperature  and 
given  out  only  at  another  temperature,  and  not  at  all  at 
intermediate  temperatures.  In  order  that  a  process,  consist- 
ing, like  those  described  above,  of  a  definite  change  in  the 
state  of  a  system  at  one  temperature  and  the  reverse  change 
in  its  state  at  another  temperature,  may  fulfil  this  last  con- 
dition, it  is  evidently  necessary  that  the  heat-capacity  of  the 
system  be  the  same  in  the  two  different  states  in  which  it 
exists  during  the  cooling  and  during  the  heating.  While  this 
is  almost  exactly  true  for  different  volumes  of  the  same 
quantity  of  any  gas  under  moderate  pressure,  and  for  the 
different  states  of  some  other  kinds  of  systems,  it  is  not  true 
in  general.  To  cyclical  processes  carried  out  with  systems 
for  which  it  is  not  true,  and  consisting  of  changes  of  the  kind 
just  specified,  the  above  equation  is  not  rigidly  applicable, 
though  it  will  give,  if  applied  to  such  processes,  approximately 
correct  results  when  the  difference  in  the  heat-capacities  of 
the  system  in  its  two  states  is  small,  or  when  the  temperature- 
difference  involved  is  small. 

It  is  evident,  that,  if  the  difference  in  the  two  tempera- 
tures be  made  infinitesimal,  the  heat-effects  involved  in  the 
two  parts  of  the  process  in  which  the  system  is  cooled 
and  heated,  whether  or  not  equal,  will  be  infinitesimal,  so 
that  they  can  be  neglected  in  comparison  with  the  finite 
quantities  of  heat  involved  in  the  other  two  parts  of  the 
process.  Therefore,  if  a  reversible  change  of  state  takes 
place  in  any  system  whatever  whereby  a  quantity  of  heat  Q 
is  absorbed  from  the  surroundings  at  the  temperature  T,  and 
a  quantity  of  work  W  is  produced  in  them,  and  the  same 
change  in  state,  but  in  the  opposite  direction,  takes  place  at  a 
slightly  different  temperature  T  +  dT,  and  thereby  a  quantity 
of  work  equal  to  —  (  W  +  d  W)  is  produced  in  the  surround- 
ings, between  these  quantities  of  work  and  heat  must  obvi- 


GENERAL  PRINCIPLES  RELATING    TO   ENERGY.         159 

ously  exist  a  relation  entirely  similar  to  that  expressed  by 
the  equation,  W^+  W2=Ql  (7\  —  T2)  /  Tlf  and  one  deriv- 
able from  it  by  substitution  ;  namely,  since  in  this  case, 
W,=  W9  W2  =  -(W+dW),  Q1  =  Q,  T,=  Tt  and 
T2  =  T  -f  dT,  it  follows  that  : 

n 
^LdT. 

The  quantity  of  heat  Q  involved  in  this  equation  is  that 
which  is  absorbed  by  the  system  when  the  change  in  it  takes 
place  under  reversible  conditions.  Since  it  is  often  not 
possible  to  measure  this  quantity  of  heat  directly,  it  is  con- 
venient to  substitute  for  it  the  quantity  W-\-(U^  —  £/i), 
which  according  to  the  First  Law  (§27)  is  equivalent  to  it. 
On  making  this  substitution  the  above  equation  becomes  : 


This  equation  is  one  of  the  most  generally  applicable  expres- 
sions of  the  Second  Law,  and  the  exact  significance  of  the 
symbols  occurring  in  it  should  be  fully  appreciated.  It  is 
evident  from  the  statements  made  in  the  foregoing  deduction 
of  the  equation  that  W  signifies  the  external  work  produced 
by  any  reversible  change  taking  place  in  any  system  at  a 
constant  temperature  T  ;  that  Uz  —  U\  is  the  attendant 
increase  in  energy  of  the  system  itself,  or  the  heat  absorbed 
when  the  same  change  in  it  takes  place  without  any  external 
work  being  done  ;  and  that  dW'is  the  increase  in  the  quantity 
of  work  produced  when  the  same  change  in  the  system  takes 
place  at  the  temperature  T  '  +  dT.  And  by  the  statement 
that  the  same  change  in  the  system  is  to  take  place  at  the 
two  temperatures,  it  is  to  be  understood  that  it  changes  from 
two  such  initial  states  as  are  converted  into  each  other  by  the 
addition  or  withdrawal  of  heat  alone,  without  any  work  being 
involved,  to  two  such  final  states  as  are  converted  into  each 
other  in  the  same  manner. 

Since  a  change  to  be  reversible  must  take  place  under 
the  equilibrium-conditions,  and  since  the    quantity  of  work 


160         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

produced  by  it  is  dependent  upon  those  conditions,  the  above 
equation  makes  it  possible  to  determine  the  effect  of  temper- 
ature on  an  equilibrium  (by  calculating  dW /  dT),  when  the 
other  quantities  (  Wand  f72  —  £7i)  involved  in  the  equation  are 
known.  The  quantity  £72  —  #i  can  usually  be  readily  deter- 
mined by  direct  calorimetric  measurements.  In  order  to 
determine  the  quantity  of  work  Wy  it  is  necessary  to  know 
the  conditions  under  which  the  change  in  state  will  take  place 
reversibly,  that  is,  to  know  the  value  of  the  energy-intensity 
within  the  system  (the  surface-tension,  pressure,  elastic  force, 
electromotive  force,  etc.)  that  must  be  compensated  by  an 
externally  applied  intensity  in  order  to  produce  equilibrium, 
and  thereby  reversibility.  The  product  of  this  intensity-value 
by  the  change  in  the  value  of  the  corresponding  capacity- 
factor  (the  change  in  surface,  volume,  length,  quantity  of 
electricity,  etc.)  is  equal  to  Wt  the  maximum  quantity  of  work 
that  can  be  produced  by  the  change  in  state  under  consider- 
ation. If  the  intensity-value  varies  during  the  change  in 
state  (as  does  the  pressure  of  a  gas  upon  expansion),  the 
maximum  work  is  given  by  a  corresponding  integral  (^or  ex- 
ample, in  the  case  of  the  gas,  by  \*pdv},  and  to  evaluate 

c/fl 

this  integral,  it  is  necessary  to  know  the  law  of  the  variation 
of  the  intensity  during  the  change,  that  is,  the  functional 
relation  between  the  intensity  and  capacity  values  (thus, 
between  /  and  v  in  the  case  of  the  gas).  All  this  know- 
ledge must  be  derived,  independently  of  the  two  Laws  of 
Energetics,  from  established  principles  relating  to  the  system 
in  question,  or  through  direct  experimental  investigations 
upon  it. 

The  way  in  which  the  equation  is  applied  in  determining 
the  effect  of  temperature  on  equilibrium-conditions  may  be 
illustrated  by  some  examples. 

Suppose  it  be  desired  to  determine  how  the  pressure  of 
a  quantity  of  hydrogen  gas  of  constant  volume  changes  with 
the  temperature.  Consider  that  the  gas  undergoes  a  change 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         1C1 

of  volume  dv  at  each  of  two  constant  temperatures  T  and 
T  +  dT>  the  pressures  of  the  gas  in  the  two  cases  being  / 
and  /  +  dp,  respectively.  Then  £72  —  £/i  =  o  by  §  37  ;  W 
=.pdv  ;  and  dW=-  dp  dv.  Substituting  in  the  general  equa- 

tion, we  get  : 

dp 


which  states  that  the  fractional  change  in  the  pressure  of  the 
gas  when  its  volume  is  kept  constant  is  equal  to  the  fractional 
change  in  its  absolute  temperature,  which  is  the  basis  of 
the  determination  of  absolute  temperature  by  the  hydrogen 
thermometer. 

As  a  second  example,  the  hydrogen  cell  may  be  considered 
for  which  the  relation  between  electromotive  force  and  tem- 
perature was  derived  above  by  means  of  a  cyclical  process. 
The  same  relation  can  be  deduced  much  more  simply  by  sub- 
stituting in  the  general  equation  the  appropriate  values  ; 
namely,  putting  IV  =  EQ,  U*  —  £/i  =  o,  dW=  Qdst  we  get  : 


In  order  to  determine  the  change  in  vapor-pressure  of  a 
pure  liquid  with  the  temperature,  we  consider  that  in  a 
system  consisting  of  the  liquid  and  its  vapor  one  mol  of  the 
liquid  changes  to  vapor  under  the  equilibrium-conditions  at 
two  different  temperatures,  T  and  T  +  dT.  Let  the  vapor- 
pressure  (that  is,  the  pressure  at  which  the  liquid  and  vapor 
are  in  equilibrium)  be  /  and  /  -f-  dp  at  the  two  temperatures, 
respectively,  and  let  the  volume  of  one  mol  of  the  liquid  be 
V,  and  that  of  one  mol  of  the  vapor  be  v,  when  the  pressure 
is  /.  Then  W=p(v  —  F)  ;  W+  £/2  —  £/i  =  ML,  the  heat 
absorbed  by  the  vaporization  of  one  mol  of  liquid  under  the 
constant  pressure  /,  which  is  the  condition  under  which  the 
calorimetric  measurements  are  ordinarily  made;  and 
dp(v  —  F).  Substituting  these  values  we  get: 

ML  =  T(v  — 
-  v 


162         GENERAL  PRINCIPLES  OF  PHYSICAL  SCIENCE. 

This  equation  evidently  gives  a  rigidly  exact  relation  between 
the  change  of  vapor-pressure  of  a  substance  with  the  tempera- 
ture, its  molar  heat  of  vaporization,  and  its  molar  volumes  in 
the  state  of  liquid  and  saturated  vapor.  In  this  equation,  if 
the  vapor-pressure  is  small  (for  example,  not  much  greater 
than  one  atmosphere)  the  volume  of  the  liquid  V  can  be 
neglected  without  great  error  in  comparison  with  that  of  the 
vapor  v9  and  for  the  latter,  as  an  approximation,  its  value 
given  by  the  gas-law-equation  (v  —  RT/p,  since  JV=  i) 
can  be  substituted.  The  above  equation  then  becomes  : 
dp  I p  __  dlvgp  ML 
~df  ''  dT  ''  ~RT* 

The  significance  of  this  equation  may  be  further  illus- 
trated, and  its  degree  of  accuracy  demonstrated,  by  applying 
it  to  water  at  100°.  The  specific  heat  of  vaporization  L  of 
water  at  100°  is  537  calories,  and  its  molecular  weight  is  18. 
Therefore  ML  =  9670  calories.  Moreover,  when  expressed 
in  calories,  R  =  1.986  (§  30) ;  and  in  this  case,  7^=373. 
Substituting  these  values  in  the  foregoing  equation,  it  is 
found  that  (dp  //)  /  dT=  0.0350.  Actual  measurements  show 
that  the  vapor-pressure  of  water  in  centimeters  of  mercury  is 
73.32  at  99°,  76.00  at  100°,  and  78.77  at  101°  ;  so  that  the 
value  of  (dp  / p)  /  dT,  is  2.725  /  76.00,  or  0.0359.  The  slight 
disagreement  between  this  experimentally  determined  value 
and  the  calculated  one  is  explained  by  the  fact  that  water- 
vapor  at  1 00°  and  atmospheric  pressure  is  not  a  perfect  gas 
and  does  not  fully  conform  to  the  equation  pv=-  NR  T,  as 
was  assumed  to  be  the  case  in  simplifying  the  exact  relation 
first  obtained. 

Since  the  fundamental  equation,  dW/dT=QI  T  = 
( W  +  U2  —  £/i)  /  T9  is  a  differential  equation,  it  is,  strictly 
speaking,  only  possible  to  deduce  from  it  the  effect  of  infini- 
tesimal changes  of  temperature  on  the  equilibrium  of  systems. 
The  effect  of  finite  changes  can,  however,  be  determined  when- 
ever this  equation  can  be  integrated  between  the  desired  tern- 
perature-limits.  But,  in  order  to  do  this,  experimental  data 


GENERAL  PRINCIPLES  RELATING    TO  ENERGY.         163 

must  be  available,  which  enable  Q,  the  heat-effect  attending 
the  reversible  change  in  state  of  the  system,  to  be  expressed 
as  a  function  of  the  temperature  between  those  limits. 

Thus,  in  the  example  last  considered,  if  it  were  desired 
to  calculate  the  vapor-pressure  of  water  at  30°  from  its  value 
at  1  00°,  it  would  be  necessary  to  know  the  change  of  the 
heat  of  vaporization  of  water  with  the  temperature.  Accord- 
ing to  Regnault's  determinations,  this  is  expressed  between 
those  temperatures  by  the  equation,  ML  =14330  —  12.5  1  T. 
Dividing  this  value  by  that  of  R,  substituting  the  result  in 
the  equation,  d\ogfl  =  MLdT/RT*,  and  integrating  be- 
tween the  limits,  T=  7i,/=/i,  and  T=  TZf  p=p*,  we  get  : 


log  =   72I8  -       r      -  6.30   log          . 

A  \yi       A/  /\ 

Placing  in  this  equation  /2  =  76.00,  T*  =  373,  and  7\  =  303, 
noting  that  natural  logarithms  are  to  be  employed,  and 
solving,  pi  is  found  to  be  3.23  cm.  Actual  measurements 
of  the  vapor-pressure  at  30°  show  it  to  be  3.15  cm. 

In  concluding  this  discussion,  attention  may  be  called 
to  the  fact  that  the  Second-Law  equation  can  also  be  applied, 
in  the  opposite  way,  to  the  determination  of  the  heat-effect 
that  attends  an  isothermal  reversible  change  in  a  system, 
when  the  effect  of  temperature  on  its  equilibrium  is  known. 
Thus,  the  heat  of  vaporization  of  a  liquid  can  evidently  be 
calculated  by  the  same  equation  as  was  used  above  in  deriv- 
ing the  temperature-coefficient  of  its  vapor-pressure,  if  the 
latter  quantity  has  been  experimentally  determined. 


APPENDIX  I. 


REFERENCES. 

§  1.    MACH,  Science  of  Mechanics,  English  Translation,  481-494. 

1893.     ("  The  Economy  of  Science. ' ' ) 
§2.    NEBNST,  Theoretische  Chemie,  1-4.    1900.   (Methods  of  science. 

Hypotheses.) 

OSTWALD,  Anorganische  Chemie,  153-155.   1900.  (Hypotheses.) 
§3.    FISKE,  Cosmic  Philosophy,  1,  215-222.     1874.     (Classification 

of  the  sciences.) 
§5.    BENO!T,  Eapports  presentes    au    Congr'es   International   de 

Physique,  1 ,  47-77.     1900.     (Meter-prototypes.) 
GUILLAUME,  ibid.,  1,  82-83.    1900     (Definition  of  liter.) 
§6.    HOLM  AN,  Matter,  Energy,  Force,  and   Work,  3-11;    18-24. 

1898.     (Presentation  of  concepts  of  matter  and  energy.) 
§7.     HOLM  AN,  Matter,  Energy,  Force,  and  Work,  142-155.     1898. 

(Properties  and  measurement  of  matter.) 
MAC&  DE  L^PINAY,  Franges  d'  interference,  Scientia  Series, 

99-101.  1902.    GUILLAUME,  Eapports  cited  above,  1,  96-99. 

1900.  (Mass  of  ccm.  of  water.) 

GUILLAUME,  ibid.,  1,80-81.     1900.     (Definition  of  kilogram.) 

LANDOLT,  Ztschr.  phys.  Chem.,  12,  1-34.    1893.    SANDFOBD 

AND  RAY,  Phys.  Rev.,  5,  247-253.    1897.     HEYDWEILLER, 

Drude's  Ann.   Phys.    (4),   5,    394-420.     1901.     (Change  of 

weight  in  chemical  reactions.) 

§§  12-14.    OSTWALD,  Lehrbuch  der  allgemeinen  Chemie,  1 ,  6-15.    1891. 
§  14.    STAS,    Nowoelles    recherches    sur    les    lois   des  proportions 
chimiques,  60-109.     1865.     (Tests   of    Law  of  Combining 
Weights.) 
§  15.    MALLET,  Memorial  Lectures  before  the  Chemical  Society,  1-56. 

1901.  (Stas.    General  discussion  of  combining-weight  de- 
terminations.) 

STAS,  Nouvelles  recherches  cited  above,  119-305.     1865. 
OSTWALD,  Lehrbuch  der  allgemeinen  Chemie,  1 ,  18-125.    1891. 
CLARKE,  Recalculation  of  the  Atomic  Weights,  1-370.     1897. 

Also,  J.  Am.   Chem.  Soc.,  19,  359-369;  20,  163-173;  21, 

200-214 ;   22,  70-80  ;  23,  90-95  ;   24,  201-215.     1897-1902. 

(Annual  reports  on  atomic-weight  determinations.) 
§16.     RICHARDS,  Proc.  Am.    Acad.     36,   544.     1901.     (Table  of 

atomic  weights.) 


REFERENCES.  165 

§21.    BOYLE,  AMAG AT,    Laws  of  Gases,  Harper's  Scientific  Series, 

1-107. 

BABUS,  ibid.,  108-110.     1899.     (Bibliography.) 
WINKELMANN,    Handbuch   der    Physik,     1,    503-523.     1891. 
REGNAULT,  Memoires  de  V Academic  des  Sciences,    26,   260. 
1893.     (Deviations  from  Boyle's  Law.     Data  f or  p&i  /  ptvz 
for  various  gases.) 

BAYNES,  Nature,  23,  186.    1880.     (pv  values  for  ethylene.) 
GUILLAUME,  Eapports  cited  above,  1 ,  82.     1900.     (Definitions 

of  atmosphere  and  normal  intensity  of  gravity.) 
§22.    GAY-LUSSAC,  REGNAULT,  et  al.,  Das  Ausdehnungsgesetz  der 

Gase,  Ostwald's  Klassiker,  1-211. 

CHAPPUIS,   Eapports  above    cited,   132-134.     1900.     (Normal 
temperature-scale.     Value  273.0°.)     Ibid.,  134-147.     (Com- 
parison with  other  scales.)     Guillaume's  Thermometrie  de 
precision,  246, 253, 256.   1889.  (Data  for  lOOcx  for  N2,  CO2,  H2.) 
REGNAULT,  Landolt  and  Bdrnsteiri's  Tabellen,  110.     (Data  for 

100*  for  CO  and  SO2.) 
§  23.    GAY-LUSSAC,  Foundations  of  the  Molecular  Theory,  Alembic 

Club  Reprints,  8-24.     (Law  of  Combining  Volumes.) 
MOBLEY,  Smithsonian  Contributions  to  Knowledge,  980,  95. 
1895.    SCOTT,  Philos.  Trans.,  1893,  543-567.     (Volumetric 
Composition  of  Water.) 
§24.    MOBLEY,  Smithsonian  Contributions  to  Knowledge,  980,  55. 

1895.     (Density  of  oxygen  at  0°.) 
§25.    HOLMAN,  Matter,  Energy,  Force,  and   Work,  25-39.    1898. 

(Forms  of  energy.) 

§26.    GBIFFITHS,    Thermal  Measurement   of  Energy,    110.     1901. 
Also  Eapports  cited  above,  1 ,  214-227.    1900.    (Standard 
calorie  and  mechanical  equivalent.) 
§  28.    OSTWALD,  Lehrbuch  der  allgemeinen  Chemie,  2, 44-50  ;  485-490. 

1893.     (Factors  of  Energy.) 
§29.    OSTWALD,  Lehrbuch  der  allgemeinen  Chemie,  2,  18-22.    1893. 

(Definitions  of  force.) 
GUILLAUME,  BOURGEOIS,  Eapports  cited  above,  1 ,  88  ;  3,  359- 

370.     1900.     (Values  of  g.) 
§  32.    SILVANUS  THOMPSON,  Electricity  and  Magnetism,  265-267 ; 

344-348  ;  586-591.     1901.     (Electrical  Units.) 
FABADAY,  Fundamental  Laws   of  Electrolytic    Conduction, 

Harper's  Scientific  Memoirs,  11-44. 
GOODWIN,  ibid.,  94-95.     1899.     (Bibliography  of  Faraday's 

Law  and  electrochemical  constant.) 

RICHABDS,  COLLINS,  AND  HEIMBOD,  Proc.  Am.  Acad.,  35, 
123-150.  1899.  (Exact  confirmation  of  Faraday's  Law,  and 
value  for  electrochemical  constant.) 

COBNU,  ABBAHAM,  Eapports  above  cited,  2,  246,  267.  1900. 
(Velocity  of  light  and  electromagnetic  waves.) 


166  NOTATION. 

§  37.  GAY-LUSSAC,  JOULE,  JOULE  &  THOMSON,  The  Free  Expansion 
of  Gases.  Harper's  Scientific  Memoirs.  1-102. 

AMES,  ibid.,  103.     1899.     (Bibliography.) 

CARNOT,  CLAUSIUS,  THOMSON.  The  Second  Law  of  Thermo- 
dynamics. Harper's  Scientific  Memoirs,  1-148. 

MAGIE,  ibid.,  149-150.     (Bibliography.) 

PLANCK,  Thermochemie,  102-106;  141-149.  1893.  (General 
Discussion.) 

NEBNST,  Theoretische  Chemie,  15-25.  1900.  (General  Discus- 
sion. Equation  dW/dT=Q/T.) 

MACH,  Warmelehre,  302-306.     1900.     (Derivation  of  equation 


LEHFELDT,  Phil.  Mag.,  45,  374-379.    1898.     (Comparison  of 
normal  and  thermodynamic  temperature-scales. ) 


APPENDIX  II. 

NOTATION. 

IN  the  system  of  notation  used  in  this  book  each  symbol 
represents,  with  a  few  unequivocal  exceptions,  only  a  single 
kind  of  physical  quantity,  and  has  in  most  of  those  cases  in 
which  usage  is  at  all  uniform,  a  significance  in  accordance 
with  that  usage.  All  physical  quantities  are  represented  by 
either  Italic  or  Greek  letters  ;  chemical  substances  and  signs 
of  operation  (log,  sin,  etc.),  by  Roman  letters.  For  all  elec- 
trical and  magnetic  quantities,  for  time,  and  for  quantities  rela- 
ting to  chemical  equivalents,  small  capitals  are  employed. 
For  all  optical  quantities  and  for  other  quantities  infrequently 
met  with,  small  Greek  letters  are  used.  To  represent  spe- 
cific properties,  the  same  symbols  are  underlined ;  thus  v  is 
volume  in  general,  and  v  is  specific  volume.  Different  values 
of  the  same  quantity  are  distinguished  by  numerical  sub- 
scripts ;  a  unique  value  by  the  subscript  zero.  The  approximate 
values  (always  within  0.5  per  cent.)  of  fundamental  con- 
stants which  it  is  well  to  remember,  are  printed  in  bold  type ; 
for  example,  ^-=980. 


NOTATION. 


167 


The  following  list  shows  the  symbols  employed  and  their 
significance. 


a    Acceleration, 
a,  b    Van  der  Waals'  constants. 

(  Energy  capacity- factor. 

C  Concentration  in  grams. 
d    Differential. 
e    Base  of  natural  logarithms. 
g    Acceleration  due  to  gravity. 
h     Height. 

.{  Energy  intensity- factor. 
*  I  Van't  Hoff  s  coefficient. 
j    Gravitation-constant. 
k    Reaction-velocity-constant 


A 

C 
D 
E 
F 
H 
/ 
K 
L 


Atomic  weight. 
Molar  concentration. 
Density. 
Energy. 
Force. 

Heat-capacity. 

Mechanical  equivalent  of  heat. 
Equilibrium-constant. 
Heat-  effect  of  vaporization  or 
fusion. 


A  Equivalent  weight. 

c  Equivalent  concentration. 

E  Electromotive  force. 

i  Current-strength. 

K  Dielectric-constant. 

L  Conductance. 

M  Quantity  of  magnetism. 


a  (  Angle  of  rotation. 

(  Empirical  coefficient 
Q  (  Absorption-coefficient 

l  Empirical  coefficient, 
y    Surface-tension. 
e    Modulus  of  elasticity. 
6    Angle. 
t    Intensity  of  light 


/  Length  or  Distance. 

m  Mass  or  Weight. 

n  Molecule- coefficients. 

p  Pressure. 

q  Heat  of  reaction. 

r  Radius. 

s  Surface  or  Cross- section. 

/  Normal  temperature. 

u  Velocity. 

v  Volume. 

x  Fractional  part  (e.g.,   content, 
dissociation,  etc.) 


M  Molecular  weight 

N  Number  of  mols. 

P  Osmotic  pressure. 

Q  Heat-effect  in  general. 

R  Gas- constant. 

S  Solubility. 

T  Absolute  temperature. 

U  Internal  energy. 

V  Volume  of  non-gaseous  phj 

W  Work. 


Number  of  equivalents. 

Quantity  of  electricity. 

Resistance. 

Time. 

Velocity  of  ions. 

Electrical  potential. 


f  (  Compression-coefficient 

(  Heat-capacity-ratio. 
A    Wave-length  of  light 
fi    Micron. 

v    Index  of  refraction. 
?r    Ratio  circumference  to  diameter. 
p    Specific  refraction, 
v    Velocity  of  light 


INDEX. 


ABSOLUTE  index  of  refraction,  127 

system  of  units,  10 

temperature-scale,  62,  156 

zero,  62 
Absorption  bands,  130 

coefficient,  130 

of  light,  130 

selective,  130 
Abstract  sciences,  9 
Acceleration,  10 

due  to  gravity,  87 
Activity,  72 

unit  of,  75 

Additive  property,  30 
Aggregates,  sciences  of,  9 
Aggregation,  states  of,  19-22 
Amorphous  substance,  24 
Ampere,  as  electrical  unit,  113 
Analyzer  for  polarized  light,  128 
Anode,  115 
A  priori  reasoning,  7 
Atmosphere,  as  unit,  57 

value  in  dynes,  95 
Atomic  weight  table,  47 

BACK  electromotive  force,  in 
Becquerel  rays,  73 
Bodv,  definition,  n 
Boiling-point,  20,  21 
Boyle's  Law,  56,  57 

deviations  from,  58-60,  137 

CALORIE,  75 

mean,  75 

mechanical  equivalent  of,  75 
Capacity  factor  of  energy,  80 

for  gravitation  energy,  14,  84 

for  heat,  120 

for  kinetic  energy,  14 
Carnot's  Cyclical  Process,  153 
Cathode,  115 
Centimeter,  10 
C.  G.  S.  electrical  units,  100,  106,  no, 

"3 

system  of  units,  10 

unit  of  magnetism,  105 
Charge,  electric,  107  [ 

Chemical  changes  in  electrolysis,  115, 
117,  118 

changes  in  voltaic  cells,  116,  119 

energy,  70,  124 

equations,  49 

formulas,  49 

substance,  criteria  of,  28-32,  48 


Chemistry,  definition,  9 

Coefficient  of  absorption  of  light,  130 

of  compression,  98 

of  expansion,  62 
Cohesion  energy,  69, 88 
Colloid,  27 

Colloidal  solution,  26 
Color,  131 

Combining-weights,    arbitrary    multi* 
pies,  41 

calculation,  41 

definition,  38,  42 

determination,  42-45 

Law  of,  38-42,  48,  49 

table  of,  47 

values,  45-47 
Complete  analysis,  44 

synthesis,  44 
Composition,  percentage-,  calculation 

of,  50 

Compound  substance,  32,  33 
Compressibility  of  gases,  59 
Compression- coefficient,  98 
Concentrated  solution,  27 
Concentration,  18 

equivalent,  56 

molar,  68 

Concepts  of  science,  10 
Concrete  sciences,  9 
Conductance,  in 
Conductivity,  in 
Conductor,  no 
Conservation  of  elements,  33,  34 

of  energy,  76-79,  138 

of  matter,  18 
Coulomb,  as  electrical  unit,  113 

Law  of,  1 01,  102 
Counter  electromotive  force,  in 
Critical  temperature,  23 
Crystalline  substance,  24 
Current,  direction  of,  106 

electric,  106 

units  of,  106,  113 
Cyclical-process,  139 

Carnot's,  153 


D ALTON'S  Law,  58 
DaniellCell,  116 
Data,  definition,  4 
Decomposition,  definition,  33 
Deductive  reasoning,  7 
Definite  Proportions,  Law  of,  36 
Degree,  centigrade,  61 


168 


INDEX. 


Density,  definition,  17 

of  air  and  oxygen,  relative,  67 

of  mercury,  95 

maximum-,  of  water,  17 
Deviations  from  Boyle's  Law,  58-60 
Dextro-rotatory  substances,  128 
Dielectric,  101 

constant,  102 
Dilute  solution,  27 
Direction  of  current,  106 
Disgregation  energy,  69 
Distance  energy,  70,  84-86 
Dyne,  87 


ELASTIC  energy,  70,  88 

energy,  factors,  98 

force,  98 

limit,  98 

modulus,  98 

Electric- charges,    attraction    and    re- 
pulsion, 100-103 

non-action  on  magnetism,  105 
Electrical-energy,  70,  99,  102,  107-109 

conversion  into  other  forms,  112 

factors,  107-109 

from  chemical  changes,  78,  143 
Electricity,  concept  of,  99, 100,  102 

current  of,  106 

direction  of  flow,  109 

distribution  upon  surfaces,  103 

positive  and  negative,  100 

simultaneous  appearance  of  the 
two  kinds,  101 

unit- quantity,  100,  106,  113 
Electrochemical  constant,  119 
Electrode,  115 
Electrolysis,  114 

chemical  changes  in,  115,  117,  118 
Electrolyte,  114 
Electrolytic  conduction,  114 
Electromagnetic  radiation,  73,  125,  126 

units,  1 06,  113 

units,  ratio,  to  electrostatic,  107 
Electromotive-force,  1 10 

calculated  from  Second  Law,  146- 

148,  156,  157,  161 
Electrostatic  unit  of  electricity,  100 

units,  ratio  to  electromagnetic,  107 
Elements,  34 

conservation,  34 

names  and  symbols,  47 

number,  34 

substances  of  same  order,  35 

transformation,  6,  33,  35 
Elementary  substance,  33 
Empirical  formula,  49 

law,  5 

Emulsion,  26 
Energetics,  definition,  76 

First  Law,  76-79,  138 

Second  Law,  137-163 


Energy  and  force  compared,  86 

chemical,  70,  124 

cohesion,  69 

conservation  of,  76-79,  138 

discussion  of,  u,  13,  69,  73 

disgregation,  69 

distance,  70,  84-86 

elastic,  70,  88 

electrical,  70,  99,  102,  107-109 

factors  in  general,  79-82 

forms,  69 

gravitation,  14,  69,  83-87 

heat,  70,  120-124 

intensity,  79 

internal,  77,  124,  132 

internal  of  gases,  132-137 

kinetic,  14,  69,  82 

magnetic,  70,  no 

measurement,  74 

mechanical,  71 

potential,  71 

radiant,  73,125-132 

surface,  70,  90-93 

unit  of,  74,  75 

volume,  70,  93-98 
Equation,  chemical,  49 
Equilibrium-conditions,  79,  142 

effect  of  temperature  on,  159-160 
Equivalent,  definition,  54 

metathetical,  54 

oxidation,  54 

weights,  52-56,  117 
Erg,  74,  87 
Ether,  definition,  125 
Expansion-coefficient,  62 

values  of,  64 
Experiments  of  Gay-Lussac,  132 

of  Joule  and  Thomson,  132 

porous-plug-,  134 
External  work,  77 

FACTORS  of  energy,  79-82 
Faraday's  Law,  114-120 
Field,  electric,  103 

intensity  or  strength,  104 
First  and   Second  Laws    contrasted, 

138 

First  Law  of  Energetics,  76-79,  138 
Force,  definitions,  84-88 

elastic,  98 

or  gravitation,  85 

of  gravity,  86 

lines  of,  104 

units  of,  87 
Formula,  chemical,  49 

empirical,  49 

molecular,  67 

weight,  51 

Forms  of  energy,  69 
Fractionation  processes,  28-29 
Friction  causes  irreversibility,  145 
Fusion,  heat  of,  121 


170 


INDEX. 


GAS  constant,  66,  96 

definition,  22 

perfect  or  ideal,  58,  137 
Gases,  laws  of,  56-67,  132-137 
Gay-Lussac's  Laws,  61,  64,  132 

unit  of  force,  87 
Gram,  unit  of  mass,  17 
Gravitational  system  of  units,  87 
Gravitation  constant,  84,  85 

energy,  discussion,  14,  69,  83-88 

energy  factors,  83-88 

force  of,  85 
Gravity,  acceleration  due  to,  86,  87 

force  of,  86 

normal  intensity,  57 

specific,  18 
Grove  Cell,  116 

HEAT  attending  expansion  of  gas,  133 

capacity,  120-122 

capacity  of  gases,  133 

effect  of    reaction,  change  with 
temperature,  122-124 

energy,  70, 120-124 

of  fusion,  121 

of  reaction,  122 

of  solution,  121 

of  vaporization,  121,  161-163 

radiant,  73 

transformation    into  work,    137- 
140,  148-155 

unit  of,  75 
Hydrogen,  properties  of,  59-65,  136 

standard  of  combining  weights,  39 
Hypothesis,  7 

IDEAL  gas,  58 

Index  of  refraction,  127 

Inductive  reasoning,  5 

Insulator,  no 

Intensity- factor  of  energy,  79 

Intensity  of  field,  104 

of  radiant  energy,  126 
Interference,  125 
Internal  energy,  77 

energy  of  gases,  132-137 
Irreversible  process,  141-145 
Isomeric  substances,  48 
Isothermal  change  or  process,  140 

changes,  Second  Law  applied  to, 
139-148 

JOULE,  as  unit  of  energy,  74 
experiments  of,  132 
Heat-Effect,  112 
Law  of,  112 

KlNERGITY,  17 
Kinetic-energy,  14,  69, 82 

factors,  82 
Kirchhoffs  Law  of  Radiation,  131 


LAEVO- rotatory  substances,  128 
Latent  heat,  121 
Law,  definition,  7 

of  Boyle,  56 

of  Combining  Volumes,  64 

of  Combining-Weights,      38-42, 
48-49 

of  Conservation  of  Energy,  76-79, 
138 

of  Conservation  of  Matter,  18 

of  Coulomb,  101,  102 

of  Dalton,  58 

of  Definite  Proportions,  35-37 

of  Energetics,  First-,  76-79,  138 

of  Energetics,  Second-,  137-163 

of  Faraday,  114,  116 

of  Gay-Lussac,  61,  64,  132 

of  Gravitation,  16,  83,  84 

of  Joule,  112 

of  Kirchhoff,  131 

of  Mariotte,  56 

of  Multiple  Proportions,  37 

of  Newton,  16,  83,  84 

of  Ohm,  no 

of  Refraction,  127 

of  Temperature-Effect,  61-64 
Light,  absorption,  130 

emission,  131 

intensity,  126 

intensity,  variation  with  distance, 
126 

magnetic  rotation,  129 

polarization,  128 

refraction,  127 

rotation  of  polarization-plane,  128 

velocity,  125,  127 

wave-length,  126 
Liquid,  definition,  24 
Lines  of  force,  104 
Liter,  n 
Logic,  9 

MAGNETIC  energy,  70,  110 

field  caused  by  electric  current,  106 

force  of  current,  106 

permeability,  105 

pole,  104 

potential,  no 

rotation,  129 
Magnetism,  concept  of,  99,  104,  105 

non-action  on  electric  charges,  105 

two  kinds,  104 

unit  of,  105 
Mariotte's  Law,  56 
Mass,  14-17,  82 
Mathematics,  9 
Matter,  concept  of,  13,  15 

conservation,  18 

quantity  of,  16,  17 

unit  of,  17 

Maximum  work,  89,  141-144,  150 
Mean  calorie,  75 


INDEX. 


171 


Mechanical  energy,  71 

equivalent  of  heat,  75 
Melting-point,  19 
Metallic  conductor,  114 

substance,  25 
Metathetical  reaction,  53 
Method,  deductive,  7 

inductive,  5 

theoretical,  4 
Methods  of  science,  4-9 
Micron,  126, 

Minimum  amount  of  work,  89,  141 
Mixtures,  26,  28,  48 
Modulus  of  elasticity,  98 
Mol,  66 

Molar  concentration,  68 
Molecular  formula,  67 

weight,  definition,  65 
Momentum,  83 

Morley,  determinations  of,  65,  66 
Motion,  10 

NEGATIVE  electricity,  100 

electrode,  115 

magnetism,  104 

Newton's  law  of  gravitation,  16,  83,  84 
Nitrogen,  pressure-volume   products, 
en,  f\ri 


Non-conductor,  no 

Non- metallic  substance,  25 

Normal  concentration,  56 

conditions,  66 

intensity  of  gravity,  57 

temperature,  62 
North  magnetic  pole,  104 

OBJECT,  definition,  n 

of  science,  3,  8 
Ohm,  as  electrical  unit,  113 

Law  of,  no,  in 

mercury  equivalent,  114 

reciprocal,  114 
Oxidation  equivalent,  54 

reaction,  53 
Oxygen,  pressure-volume  data,  62 

standard  of  combining  weights,  39 


PARTIAL  pressure,  58 

Percentage  composition, calculation,  50 

Perfect  gas,  58,  137 

Permeability,  magnetic,  105 

Perpetual-motion  of  first  kind,  77 

of  second  kind,  137 
Phase,  27 
Phenomenon,  n 
Physics,  9 

Physical  Sciences,  9 
Plane  of  polarization,  1 28 

-polarized  light,  128 


Polarized  light,  128 
Pole,  magnetic,  104 

north  and  south,  104 

strength,  105 

Porous-plug  experiments,  134 
Positive  electricity,  100 

electrode,  115 

magnetism,  104 
Potential,  definition,  107 

difference,  unit  of,  108 

energy,  71 
Power,  72 

unit  of,  75 

Practical  system  of  electrical  units,  113 
Pressure,  definition,  74 

in  tensity- factor,  80 

units  of,  57,  95 

Pressure-volume  product  at  constant 
temperature,  57-60 

product,    relation    to    combining 
weight,  64,  65 

product,  relation  to  temperature, 
61-64 

relations  of  gases,  65-67 
Process,  cyclical,  139 

definition,  139 

irreversible,  141 

reversible,  141 
Property,  additive,  30 

definition,  n 

specific,  12 
Pseudo-solution,  25 


QUANTITY  of  electricity,  100 
of  heat,  expression  of,  75 
of  magnetism,  no 
of  matter,  16 


RADIANT  energy,  73,  125-132 

heat,  73,  125,  126 
Radiations,   electromagnetic,  73,   125, 

126 
Reaction,  chemical,  51 

metathetical,  53 

of  oxidation  and  reduction,  53 
Reciprocal  ohm,  114 
Reduction,  reaction,  53 
Refraction,  127 

Relative  and  absolute  densities,  63 
Resistance,  definition,  1 10 

specific,  in 

unit  of,  no,  113 
Resistivity,  in 
Reversible  change  or  process,  141 

cyclical  processes,  145-157 
Reversibility,  conditions  of,  142-145, 

149,  151 

Rontgen  Rays,  73 
Rotatory  power,  129 


172 


INDEX. 


SCIENCE,  fundamental  concepts,  10 

methods,  4 

objects,  3 

subdivisions,  9 
Second  (time  unit),  n 
Second  Law  equations,  153,  159 

and  First  Law  contrasted,  138 

of  Energetics,  137-163 
Selective  absorption,  131 
Semipermeable  wall,  26 
Snail's  Law  of  Refraction,  127 
Solid,  definition,  24 
Solute,  27 
Solution,  25 

colloidal,  26 

pseudo-,  26 
Solvent,  27 

South  magnetic  pole,  104 
Space,  concept  of,  10 
Specific  conductance,  in 

gravity,  18 

heat  or  heat- capacity,  120 

heat  of  fusion,  vaporization,  121 

property,  12 

resistance,  in 

rotatory  power,  1 29 

volume,  18 
Spectrum,  130 

Standard  of  combining  weights,  39 
Stas,  experiments  of,  36,  38,  42-45 
States  of  aggregation,  19-22 
Strength  of  field,  104 

of  pole,  105 
Substance,  chemical,  28-32,  48 

compound,  32-33 

definition,  u,  12 

pure,  28-32,  48 
Surface  color,  131 

density,  103 

energy,  def.,  70 

energy,  factors,  88 

tension,  91 

tension,  determination  of,  91 

tension,  unit  of,  93 
Surroundings,  77 
Suspension,  26 
Symbols  of  elements,  47 
System,  absolute-,  of  units,  10 

of  bodies,  77 


TEMPERATURE  coefficient  of  calorie,  75 
critical,  23 

effect  on  pressure- volume  prod- 
uct, 61 


Temperature 

effect  on  vapor- pressure,  162 

intensity-factor  of  heat,  79 

scale,  absolute,  62,  156 

scale,     energetic     or     thermody- 
namic,  156 

scales,  hydrogen  and  perfect  gas 
compared,  155 

scale,  normal,  62 
Theoretical  principle,  7 
Theory,  7 

Thermodynamic  temperature-scale,!  56 
Thomson  and  Joule  experiments,  132 
Time,  concept  of,  10 
Torsion-balance,  102 


UNITS,  C.  G.  S.,  10,  87,  100,  106 
international  electrical,  114 
practical  electrical,  113 
systems  of,  10,  87,  100,  106,  113 
true  electrical,  114 


VAN  DER  WAALS'  equation,  59 
Vapor,  23 

Vaporization,  heat  of,  121,  161-163 
Vapor- pressure,  change  with  tempera- 
ture, 161-163 

definition,  20 
Velocity  of  bodies,  10 

of  light,  125,  127 
Volt,  113 
Voltaic  action,  115 

cells,  chemical  changes  in,  1 16, 1 19 
Volume  energy,  70,  93-98 

energy,  factors,  93-94 

specific,  18 

WATT,  as  unit,  76 
Waves  of  radiant  energy,  125 
Wave-length,  126 
Weight,  1 6,  17 

changes  in  chemical  reactions,  18 
Work,  definition,  71 

done  in  production  and  expansion 
of  gases,  95-97 

done  in  volume-changes,  93-98 

electrical,  78,  112,  143 

external,  77 

maximum,  89,  141-144,  15° 

ZERO,  absolute,  of  temperature,  62 
of  potential,  108  N 


OF  THE 

{   UNIVERSITY  } 


*- 


FOURTEEN  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 


This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 

&"'  \  ->••-  " 


'D  LD 


NOV  20  1956 


D  LD 


l-tb  k)7  19: 


REC'D  LO 


General  Library 


TO 


